Generalized Trigonometric and Hyperbolic Functions

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Generalized Trigonometric and Hyperbolic Functions

Generalized Trigonometric and Hyperbolic Functions

Ronald E. Mickens

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2019 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper Version Date: 20181221 International Standard Book Number-13: 978-1-138-33301-7 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Names: Mickens, Ronald E., 1943- author. Title: Generalized trigonometric and hyperbolic functions / Ronald E. Mickens. Description: Boca Raton, Florida : CRC Press, [2019] | Includes bibliographical references. Identifiers: LCCN 2018038380| ISBN 9781138333017 (hardback : alk. paper) | ISBN 9780429446238 (ebook). Subjects: LCSH: Trigonometry. | Exponential functions. | Hyperbola. Classification: LCC QA531 .M5845 2019 | DDC 516.24/6--dc23 LC record available at https://lccn.loc.gov/2018038380

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Dedication

This book is dedicated to my wife Maria, whose actions provided the family and home that I needed . . . and, as one of its consequences, made me a better person!

Contents

Dedication

v

List of Figures

xi

Preface

xiii

Author

xvii

1 TRIGONOMETRIC AND HYPERBOLIC SINE AND COSINE FUNCTIONS 1.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 1.2 SINE AND COSINE: GEOMETRIC DEFINITIONS . . . . 1.3 SINE AND COSINE: ANALYTIC DEFINITION . . . . . . 1.3.1 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Addition and Subtraction Rules . . . . . . . . . . . . . 1.3.5 Product Rules . . . . . . . . . . . . . . . . . . . . . . 1.4 SINE AND COSINE: DYNAMIC SYSTEM APPROACH . . 1.4.1 x-y Phase-Space . . . . . . . . . . . . . . . . . . . . . 1.4.2 Symmetry Properties of Trajectories in Phase-Space . 1.4.3 Null-Clines . . . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Geometric Proof that All Trajectories Are Closed . . . 1.5 HYPERBOLIC SINE AND COSINE: DERIVED FROM SINE AND COSINE . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 HYPERBOLIC FUNCTIONS: DYNAMIC SYSTEM DERIVATION . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 θ-PERIODIC HYPERBOLIC FUNCTIONS . . . . . . . . . 1.8 DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . . Notes and References 2 ELLIPTIC FUNCTIONS 2.1 INTRODUCTION . . . . . . . . . . . . . . 2.2 θ-PERIODIC ELLIPTIC FUNCTIONS . . 2.3 ELLIPTIC HAMILTONIAN DYNAMICS . 2.4 JACOBI, CN, SN, AND DN FUNCTIONS

1 1 2 3 5 6 6 7 7 8 9 10 10 13 14 17 18 20 22

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23 23 23 26 28 vii

viii

Contents

2.5

2.6

2.7 2.8 2.9

2.4.1 Elementary Properties of Jacobi Elliptic Functions . . 2.4.2 First Derivatives . . . . . . . . . . . . . . . . . . . . . 2.4.3 Differential Equations . . . . . . . . . . . . . . . . . . 2.4.4 Calculation of u(θ) and the Period for cn, sn, dn . . . 2.4.5 Special Values of Jacobi Elliptic Functions . . . . . . . ADDITIONAL PROPERTIES OF JACOBI ELLIPTIC FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Fundamental Relations for Square of Functions . . . . 2.5.2 Addition Theorems . . . . . . . . . . . . . . . . . . . . 2.5.3 Product Relations . . . . . . . . . . . . . . . . . . . . 2.5.4 cn, sn, dn for Special k Values . . . . . . . . . . . . . 2.5.5 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . DYNAMICAL SYSTEM INTERPRETATION OF ELLIPTIC JACOBI FUNCTIONS . . . . . . . . . . . . . . . . . . . . . 2.6.1 Definition of the Dynamic System . . . . . . . . . . . 2.6.2 Limits k → 0+ and k → 1− . . . . . . . . . . . . . . . 2.6.3 First Integrals . . . . . . . . . . . . . . . . . . . . . . 2.6.4 Bounds and Symmetries . . . . . . . . . . . . . . . . . 2.6.5 Second-Order Differential Equations . . . . . . . . . . 2.6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . HYPERBOLIC ELLIPTIC FUNCTIONS AS A DYNAMIC SYSTEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HYPERBOLIC θ-PERIODIC ELLIPTIC FUNCTIONS . . . DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . .

29 30 31 32 33 34 34 34 36 36 36 37 37 38 38 39 40 40 40 41 45

Notes and References

45

3 SQUARE FUNCTIONS 3.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 3.2 PROPERTIES OF THE SQUARE TRIGONOMETRIC FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 PERIOD OF THE SQUARE TRIGONOMETRIC FUNCTIONS IN THE VARIABLE . . . . . . . . . . . . . . 3.4 FOURIER SERIES OF THE SQUARE TRIGONOMETRIC FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 DYNAMIC SYSTEM INTERPRETATION OF |x| + |y| = 1 3.6 HYPERBOLIC SQUARE FUNCTIONS: DYNAMICS SYSTEM APPROACH . . . . . . . . . . . . . . . . . . . . . 3.7 PERIODIC HYPERBOLIC SQUARE FUNCTIONS . . . .

47 47

Notes and References

65

4 PARABOLIC TRIGONOMETRIC FUNCTIONS 4.1 INTRODUCTION
. . . . . . . . . . . . . . . . . . . . . . . . 4.2 H(x, y) = |y| + 12 x2 AS A DYNAMIC SYSTEM . . . . . .

67 67 67

50 51 53 56 59 64

Contents 4.3 4.4 4.5

ix


GEOMETRIC ANALYSIS OF |y| + 12 x2 = 12 . . . . . . .
|y| − 12 x2 = 12 AS A DYNAMIC SYSTEM . . . . . . . . .
GEOMETRIC ANALYSIS OF |y| − 12 x2 = 12 . . . . . . .

Notes and References

5 GENERALIZED PERIODIC SOLUTIONS OF f (t)2 + g(t)2 = 1 5.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . 5.2 GENERALIZED COSINE AND SINE FUNCTIONS 5.3 MATHEMATICAL STRUCTURE
OF θ(t) . . . . . 5.4 AN EXAMPLE: A(t) = a1 sin 2π T t . . . . . . . . . 5.5 DIFFERENTIAL EQUATION FOR f (t) AND g(t) 5.6 DISCUSSION . . . . . . . . . . . . . . . . . . . . . 5.7 NON-PERIODIC SOLUTIONS OF f 2 (t) + g 2 (t) = 1

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Notes and References

72 73 78 81 83 83 84 86 88 89 92 94 97

6 RESUMÉ OF (SOME) PREVIOUS RESULTS ON GENERALIZED TRIGONOMETRIC FUNCTIONS 6.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 6.2 DIFFERENTIAL EQUATION FORMULATION . . . . . . 6.3 DEFINITION AS INTEGRAL FORMS . . . . . . . . . . . . 6.4 GEOMETRIC APPROACH . . . . . . . . . . . . . . . . . . 6.5 SYMMETRY CONSIDERATIONS AND CONSEQUENCES 6.5.1 Symmetry Transformation and Consequences . . . . . 6.5.2 Hamiltonian Formulation . . . . . . . . . . . . . . . . 6.5.3 Area of Enclosed Curve . . . . . . . . . . . . . . . . . 6.5.4 Period . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . .

99 99 100 101 102 103 103 104 105 106 107

Notes and References

107

7 GENERALIZED TRIGONOMETRIC FUNCTIONS: |y|p + |x|q = 1 7.1 INTRODUCTION . . . . . . . . . . . . . . . 7.2 METHODOLOGY . . . . . . . . . . . . . . 7.3 SUMMARY . . . . . . . . . . . . . . . . . . 7.4 GALLERY OF PARTICULAR SOLUTIONS

. . . .

109 109 109 111 111

8 GENERALIZED TRIGONOMETRIC HYPERBOLIC FUNCTIONS: |y|p − |x|q = 1 8.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 8.2 SOLUTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 GALLERY OF SPECIAL SOLUTIONS . . . . . . . . . . . .

119 119 120 120

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x

Contents

9 APPLICATIONS AND ADVANCED TOPICS 127 9.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 127 9.2 ODD-PARITY SYSTEMS AND THEIR FOURIER REPRESENTATIONS . . . . . . . . . . . . . . . . . . . . . 130 9.3 TRULY NONLINEAR OSCILLATORS . . . . . . . . . . . . 133 9.3.1 Antisymmetric, Constant Force Oscillator . . . . . . . 134 9.3.2 Particle in a Box . . . . . . . . . . . . . . . . . . . . . 136 9.3.3 Restricted Duffing Equation . . . . . . . . . . . . . . . 137 9.4 ATEB PERIODIC FUNCTIONS . . . . . . . . . . . . . . . 138 9.5 EXACT DISCRETIZATION OF THE JACOBI ELLIPTIC DIFFERENTIAL EQUATIONS . . . . . . . . . . . . . . . . 140 9.5.1 Rescaled Duffing Equation . . . . . . . . . . . . . . . . 140 9.5.2 Exact Difference Equation for CN . . . . . . . . . . . 141 9.5.3 Exact Difference Equation for SN . . . . . . . . . . . . 142 9.6 HARMONIC BALANCE: DIRECT METHOD . . . . . . . . 143 9.6.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . 143 9.6.2 x ¨ + x3 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . 145 9.6.3 x ¨ + x1/3 = 0 . . . . . . . . . . . . . . . . . . . . . . . 147 9.7 HARMONIC BALANCE: RATIONAL APPROXIMATION 149 9.7.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . 149 9.7.2 x ¨ + x3 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . 150 9.7.3 x ¨ + x2 sgn(x) = 0 . . . . . . . . . . . . . . . . . . . . . 152 9.8 ITERATION METHODS . . . . . . . . . . . . . . . . . . . . 153 9.8.1 Direct Iteration Scheme: x¨ + x3 = 0 . . . . . . . . . . 155 9.8.2 Extended Iteration: x¨ + x3 = 0 . . . . . . . . . . . . . 157 9.9 DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Notes and References

161

10 FINALE 10.1 GOALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 SOME UNRESOLVED TOPICS AND ISSUES . . . . . . .

165 165 165 166

Notes and References

168

APPENDIX

171

Bibliography

185

Index

189

List of Figures

1.2.1 Plot of circle x2 + y 2 = 1 and the definitions of cos θ and sin θ, respectively as x/r and y/r. . . . . . . . . . . . . . . . . . . 1.2.2 Plots of cos θ and sin θ vs θ. . . . . . . . . . . . . . . . . . . 1.4.1 Symmetry transformations, in the x-y phase-plane. . . . . . 1.4.2 Symmetry transformations for segments of curves. . . . . . . 1.4.3 Null-clines and signs of dy/dx in the four quadrants for the harmonic oscillator dynamic system. . . . . . . . . . . . . . 1.4.4 Geometric proof of simple, closed phase-space trajectories for the harmonic oscillator dynamic system. . . . . . . . . . . . 1.5.1 The hyperbolic cosine and sine functions. . . . . . . . . . . . 1.6.1 Plot of y 2 − x2 = 1. . . . . . . . . . . . . . . . . . . . . . . . 1.7.1 Definition of the variables r and θ. . . . . . . . . . . . . . . 1.7.2 Plots of r(θ) and x(θ) from Equations (1.86). . . . . . . . .

2 4 11 12 14 15 16 18 19 21

2

2.2.1 Graph of y 2 + xa2 = 1, and the definitions of r and θ. P (x, y) denotes a point on the ellipse, and a > 1. . . . . . . . . . . . 2.2.2 Graphs of edn(θ), ecn(θ), and esn(θ) for a = 23 or k 2 = 59 . . 2.4.1 Graphs of the three Jacobi elliptic functions for k = 0.8. . . 2.7.1 Graphs of Equations (2.112). . . . . . . . . . . . . . . . . . . 2.7.2 Graphs of Equations (2.113). . . . . . . . . . . . . . . . . . .
2 2.8.1 Graphs of y 2 − xa = 1, and definition of r, θ, and P (x, y). 3.1.1 3.1.2 3.5.1 3.6.1 3.6.2

3.7.1 4.1.1 4.2.1 4.3.1

(a) Plot of |x| + |y| = 1. (b) Definition of r and θ. . . . . . . Plots of sqd(θ), sqc(θ), and sqs(θ) vs θ. . . . . . . . . . . . . Plots of x(t) and y(t) vs t. See Equations (3.59). . . . . . . Plot of H(x, y) = |y| − |x| = 1. . . . . . . . . . . . . . . . . The flow along the trajectories in the x-y phase-space for H(x, y) = |y| − |x| = 1. See Equations (3.65). . . . . . . . . Definition of r and θ for the periodic hyperbolic square functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Plot of |y| + 21 x2 = 12 . . . . . . . . . . . . . . . . . . . . . Plots of x(t) and y(t) vs t. See Equations (4.9), (4.15), (4.17) and (4.18). . . . . . . . . . . . . . . . . . .
. . . . . . . . . . Definition of r and θ for the curve |y| + 12 x2 = 12 . . . . . .

24 27 35 42 43 44 48 49 58 60 61 64 68 71 72

xi

xii

List of Figures 4.3.2 Graphs of the parabolic trigonometric functions. . . . . . . .
4.4.1 Graphs of H(x, y) = |y|− 12 x2 = 12 , along with the directions of the phase-space flow. . . . . . . . . . . .
. . . . . . . . . . 4.5.1 Definition of r and θ for the curve |y| − 12 x2 = 12 . . . . . . 5.4.1 Plots of θ(t) vs t and d(t) vs t, for a1 = 0.2. . . . . . . . . . 5.4.2 Plots of c(t) and s(t). . . . . . . . . . . . . . . . . . . . . . . 5.7.1 Two oscillatory, non-periodic functions, corresponding to θ(t) = t2 and θ(t) = t + t2 . . . . . . . . . . . . . . . . . . . . 5.7.2 Two oscillatory, non-periodic functions, corresponding to θ(t) = t2 /(1 + t2 ). . . . . . . . . . . . . . . . . . . . . . . . .

74 76 79 90 91 95 96

6.5.1 Plot of the phase-space trajectory in the fourth-quadrant beginning at (1, 0) and ending at (0, −1). The time to do this is Tpq /4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

104

7.4.1a Plot of |y|1.2 + |x|2 = 1, i.e., p = 1.2 and q = 2. 7.4.1b Plots of x1.2,2 (t) and y1.2,2 (t), versus t. . . . . . 7.4.2a Plot of |y|2 + |x|4/3 = 1, i.e., p = 2 and q = 4/3. 7.4.2b Plot of x2,4/3 (t) and y2,4/3 (t), versus t. . . . . . 7.4.3a Plot of |y|3 + |x|4 = 1, i.e., p = 3 and q = 4. . . 7.4.3b Plot of x3,4 (t) and y3,4 (t), versus t. . . . . . . .

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112 113 114 115 116 117

8.3.1a Plot of |y|1.2 − |x|2 = 1, for p = 1.2 and q = 2. . 8.3.1b Plots of y1.2,2 (t) and x1.2,2 (t), versus t. . . . . . 8.3.2a Plot of |y|2 − |x|4/3 = 1, for p = 2 and q = 4/3. 8.3.2b Plots of y2,4/3 (t) and x2,4/3 (t), versus t. . . . . 8.3.3a Plot of |y|3 − |x|4 = 1, for p = 3 and q = 4. . . . 8.3.3b Plots of y3,4 (t) and x3,4 (t), versus t. . . . . . . .

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121 122 123 124 125 126

C.2.1 Plot of the absolute value function, |x| . . . . . . . . . . . . C.3.1 (a) Plot of sign(x) vs x. (b) Plot of θ(x) vs x . . . . . . . . .

178 179

Preface

The major purpose of this book is to introduce and investigate various topics related to so-called generalized trigonometric and hyperbolic functions. The methodology and the associated analysis are essentially mine and in many cases are not connected mathematically with previous work on this subject. In general, our obtained results are derived and discussed using a “rigorous heuristic” style of mathematical analysis. However, in spite of what some might consider a limitation, this procedure allows very interesting results to follow. The background knowledge needed to study with profit and to understand the contents of the book consists of the reader having taken courses in elementary plane geometry, trigonometry, and a year of calculus. My interests in generalized trigonometric and hyperbolic functions had their genesis nearly four decades ago when I simultaneously began the study of and the writing of a book on nonlinear oscillations: • R. E. Mickens, Introduction to Nonlinear Oscillations (Cambridge University Press, New York, 1981). For many conservative one-degree-of-freedom systems, the energy function, i.e., the Hamiltonian function, takes the form H(x, y) =

y2 + x2n , 2

n = 0, 1, 2, . . . ,

(0.1)

where

dx(t) , x(0) = 1, y(0) = 0, (0.2) dt and we have used scaling to eliminate unneeded parameters, and to normalize the energy to H(x, y) = 1. A general extension of Equations (0.1) and (0.2) is y(t) ≡

H(x, y) = |y|p + |x|q = 1, x(01 = 1, p > 0,

y(0) = 0,

(0.3)

q > 0,

where p and q are real. Since in the x-y plane the solution curves, y = y(x), are simple, closed curves, then all the solutions, (x(t), y(t)), are periodic. It is important to note that in the first of Equations (0.3), the absolute value appears. This requirement is critical for our work, otherwise, the curves may not be simple and closed. xiii

xiv

Preface

Our next spark of interest occurred with the reading of a paper by Schwalm, i.e., • W. Schwalm, Elliptic functions sn, cn, dn as trigonometry: http://www.und.edu/instruct/schwalm/MAA_Presentation_10_02/ handout.pdf, 2005. This work clearly indicated the potential of using planar geometrical concepts to analyze two-dimensional dynamic systems in such a manner that new classes of periodic functions could be created. This book illustrates what can be done using this methodology. Another important point is that the results reported in this book are greatly constrained by the condition that |y|p ± |x|q = 1 is invariant under the transformations T1 : x → −x, T2 : x → x,

y→y

y → y−y

(0.4)

T3 = T1 T2 = T2 T1 : x → −x,

y → −y.

One consequence is that periodic and non-periodic solutions have many of the same general mathematical features as those possessed by the standard trigonometric and hyperbolic functions. The book consists of ten chapters, an appendix, and a short bibliography. Chapters 1 and 2 restate and reformulate the standard results known, respectively, on the trigonometric cosine and sine functions, and the associated hyperbolic functions. The techniques used for this analysis are then applied, in Chapters 3 and 4, to the “square” and “parabolic” periodic and hyperbolic functions. Our major, original contribution is to demonstrate the existence of periodic hyperbolic-type functions. Chapter 5 provides a detailed discussion of a particular class of periodic solutions to the functional equation f (t)2 + g(t)2 = 1, f (0) = 1,

g(0) = 0.

(0.5)

A rather cursory presentation is given in Chapter 6 on some of the previous work done on the so-called generalized trigonometric functions. Most of these formulations were done within the framework of integral relations. We provide references to publications which are of direct relevance to the general topics presented in this book. Chapters 7 and 8, while the shortest chapters, are the central features of the book. They define a new class of generalized trigonometric and hyperbolic functions based on the functional equations |y|p + |x|q = 1;

x(0) = 0,

y(0) = 0;

(0.6)

Preface |y|p − |x|q = 1; p > 0,

x(0) = 0, y(0) = 1, q > 0.

xv (0.7) (0.8)

Chapter 9 gives an analysis of several cases which model important dynamic systems involving, in general, nonlinear oscillations. For these situations p = 2 and q ≥ 1. Chapter 10 lists and briefly discusses a number of unresolved issues in the field of generalized trigonometric and hyperbolic functions. The Appendix includes several mathematical listings of certain special functions and their properties, along with related materials on Fourier series, the delta and sign functions, and a brief summary of Hamiltonian dynamics. Finally, the short Bibliography gives a listing of books, publications, and websites which were helpful to the writing of this volume. As always, I am truly thankful to Annette Rohrs for her outstanding technical and editorial services, which allowed my handwritten pages to be transformed into this volume. Both she and my wife, Maria Mickens, provided valuable assistance and encouragement to successfully finish this writing project on time. I acknowledge and thank the editor, Sarfraz Khan, for his help in the initiation and production of this book. Further thanks and great appreciations are extended to my former student and now colleague, Kale Oyedeji (Morehouse College, USA), and my colleague and friend, Ivana Kovacic (University of Novi Sad, Serbia). Both have provided aid and insights that have enhanced my search for deep understandings of a range of issues in nonlinear dynamics and special functions. Finally, it should be noted that my research was especially enhanced by the efforts of two persons, both reference librarians at the Robert Woodruff Library, Atlanta University Center: Imani Beverly and Bryan Briones. They obtained needed copies of articles and books, often within days of my requests, and in addition, provided general cheerful and professional services. Ronald E. Mickens Atlanta, Georgia, USA August 2018

Author

Ronald Elbert Mickens received his BA degree in physics from Fisk University (1964) and a Ph.D. in theoretical physics from Vanderbilt University (1968). He held postdoctoral positions at the MIT Center for Theoretical Physics (1968–70), Vanderbilt University (1980–81), and the Joint Institute for Laboratory Astrophysics (1981–82). He was professor of physics at Fisk University from 1970 to 1981. Presently, he is the Distinguished Fuller E. Callaway Professor at Clark Atlanta University. His current research interests include nonlinear oscillations, asymptotic methods for difference and differential equations, numerical integration of differential equations, the mathematical modelling of periodic diseases, the history/sociology of African-Americans in science, and the relationship between mathematics and physics. As of 2016, he has published more than 335 peer-reviewed scientific/mathematical research articles; written and/or edited 17 books; published over 390 abstracts; and authored nearly 100 scientific bio-essays, book reviews, and commentaries. He serves on editorial boards of several research journals, including the Journal of Difference Equations and Applications and the International Journal of Evolution Equations. His scholarly writings have appeared in reference works such as African American Lives (Oxford University Press), American National Biography (Oxford University Press), and Biographical Encyclopedia of Scientists (Marshall Cavendish). His honors include fellowships from the Ford, Woodrow Wilson, and National Science Foundations, as well as election to Phi Beta Kappa (1964). During 1998–99, he was an American Physical Society (APS) Centennial speaker (as part of the activities to celebrate the 100th anniversary of the founding of the APS). He also served as a Distinguished National Lecturer for Sigma Xi, The Scientific Research Society for 2000–2002. His professional memberships include the American Association for the Advancement of Science, the APS (for which he is an elected Fellow), the History of Science Society, the Society for Mathematical Biology, and the American Mathematical Society. In July 2014, “The Brauer–Mickens Distinguished Seminar Series,” in the Simon A. Levin Mathematical, Computational and Modeling Sciences Center (Arizona State University), was inaugurated to honor Ronald Mickens for his “stellar scholarly contributions to the mathematical, engineering, and natural sciences . . . and (his) overall service and mentorship to the applied mathematical sciences community.”

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Author

In 2018, Ronald E. Mickens won the Blackwell-Tapia Prize, recognizing him as a mathematician who has contributed significantly to research in applied mathematics and who has been a role model for mathematical scientists and students from underrepresented groups. Access to a multi-hour interview with Professor Mickens is posted at www.thehistorymakers.org/biography/ronald-mickens-41. This interview covers a variety of issues related to his family life, career, and scientific contributions. The Amistad Research Center, Tulane University, New Orleans, LA, houses a large collection of his personal and scientific correspondence.

Chapter 1 TRIGONOMETRIC AND HYPERBOLIC SINE AND COSINE FUNCTIONS

1.1

INTRODUCTION

Historically, the trigonometric sine and cosine functions had their genesis in the study of properties of the magnitude of line segments contained within the perimeter of a circle. This chapter examines these functions from several different perspectives and ends with the introduction of a new set of functions which we denote the θ-periodic functions. Section 1.2 starts with the standard definition of the sine/cosine functions based on the geometric properties of the circle. Section 1.3 is based on an analytic definition of sine/cosine which follows from the Euler relation eiθ = cos θ + i sin θ.

(1.1)

Another interesting and important way of examining the properties of sine/cosine functions is to consider them solutions to a dynamic system modeling the simple harmonic oscillator. These results are presented in Section 1.4. The next three sections are devoted to the associated hyperbolic functions. In the simplest, geometric case, this is accomplished by replacing θ by iθ, where √ i = −1. This is done in Section 1.5. The hyperbolic functions can also be examined by considering them to be solutions to a particular dynamic system, and these considerations are given in Section 1.6. While the usual, standard trigonometric hyperbolic functions are not periodic, we show in Section 1.7 how to construct periodic generalizations; we call them θ-periodic hyperbolic functions. Finally, we conclude the chapter with a summary of the obtained results, along with the inclusion of comments on what these results (might) mean.

1

2 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS

FIGURE 1.2.1: Plot of circle x2 + y 2 = 1 and the definitions of cos θ and sin θ, respectively as x/r and y/r.

1.2

SINE AND COSINE: GEOMETRIC DEFINITIONS

The unit circle, in the x-y plane, is defined by the formula x2 + y 2 = 1.

(1.2)

Let a ray be drawn from the origin O, such that it intersects the perimeter of the circle at point P (x, y); see Figure 1.2.1. Denote by θ, the angle that this ray makes with the positive x-axis. The sine and cosine functions are defined to be ( sin θ ≡ yr , (1.3) cos θ ≡ xy .

TRIGONOMETRIC & HYPERBOLIC SINE & COSINE FUNCTIONS 3 Since r = 1, for the unit circle, these definitions reduce to sin θ = y,

cos θ = x.

(1.4)

Taking into consideration the geometry in Figure 1.2.1 and further taking into consideration the fact that x and y are “signed” (positive or negative) real numbers, the following is a listing of some of the major properties of the sine and cosine functions: (cos θ)2 + (sin θ)2 = 1 cos(−θ) = cos θ, −1 ≤ |cos θ| ≤ +1,

sin(−θ) = − sin θ −1 ≤ |sin θ| ≤ +1

cos(θ + 2π) = cos θ, sin(θ + 2π) = sin θ
cos(0) = 1, cos π2 = 0, cos(π) = −1
cos 3π = 0, cos(2π) = 1 2 (
sin(0) = 0, sin π2 = 1, sin(π) = 0
sin 3π = −1, sin(2π) = 0 2 ( (+), 0 < θ < π sgn(sin θ) = (−), π < θ < 2π  π  (+), 0 < θ < 2 sgn(cos θ) = (−), π2 < θ < 3π 2   (+), 3π 2 < θ < 2π       Max |cos θ| Min |sin θ| = Min |sin θ| Max |cos θ|

(

1.3

(1.5) (1.6) (1.7) (1.8) (1.9)

(1.10)

(1.11)

(1.12)

(1.13)

SINE AND COSINE: ANALYTIC DEFINITION

An alternative way to introduce and investigate the sine and cosine functions is to start with the Euler formula eiθ = cos θ + i sin θ.

(1.14)

Note that this is the same as Equation (1.1), but for convenience, we rewrite it again, with a new numbering, for ease of reference to it in this section.

4 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS

FIGURE 1.2.2: Plots of cos θ and sin θ vs θ.

TRIGONOMETRIC & HYPERBOLIC SINE & COSINE FUNCTIONS 5 Taking the complex conjugate of Equation (1.14) and multiplying it by Equation (1.14) gives (e−iθ )(eiθ ) = (cos θ − i sin θ)(cos θ + i sin θ) 1 = (cos θ)2 + (sin θ)2 ,

(1.15)

and this is just the previously obtained result stated in Equation (1.5). Since ei(−θ) = (eiθ )∗ , (1.16) it follows that cos(−θ) + i sin(−θ) = cos θ − i sin θ,

(1.17)

and we conclude that cos(−θ) = cos θ,

sin(−θ) = − sin θ,

(1.18)

i.e., the cosine and sine functions are, respectively, even and odd functions of θ. In the next several subsections, we derive other properties of the cosine/sine functions.

1.3.1

Derivatives

Taking the derivatives of both sides of Equation (1.14) gives d iθ d cos θ d sin θ (e ) = +i . dθ dθ dθ

(1.19)

But d iθ (e ) = ieiθ dθ = − sin θ + i cos θ,

(1.20)

and comparing the corresponding real and imaginary parts of Equations (1.19) and (1.20), allows the following conclusions to be made d cos θ = − sin θ, dθ

d sin θ = cos θ. dθ

(1.21)

If now the derivative is taken of the results expressed in Equation (1.21), we obtain d sin θ d2 cos θ =− = − cos θ, dθ2 dθ 2 d cos θ d sin θ = = − sin θ. 2 dθ dθ

(1.22a) (1.22b)

6 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS Consequently, cos θ and sin θ both satisfy the following second-order, linear differential equation, with constant coefficients d2 w(θ) + w(θ) = 0. dθ2

(1.23)

Observe that the general solution for Equation (1.23) is w(θ) = A cos θ + B sin θ,

(1.24)

where A and B are arbitrary real constants. Thus, the cosine and sine functions are solutions to the following initial value problems to Equation (1.24): cos θ : w(0) = 1, sin θ : w(0) = 0,

1.3.2

dw(0) = 0, dθ dw(0) = 1. dθ

(1.25a) (1.25b)

Integrals

It follows directly from the results presented in Equation (1.2) that Z sin θ dθ = − cos θ, (1.26a) Z cos θ dθ = sin θ. (1.26b)

1.3.3

Taylor Series

The exponential function exp(u) has the Taylor series u

e =

∞ X uk

k=0

k!

.

(1.27)

Therefore, for u = iθ, we have eiθ =

∞ X (iθ)k k=0

k!

.

(1.28)

If we now use (i)0 = 1,

(i)1 = i,

(i)2 = −1,

(i)3 = −i,

(i4 ) = 1,

(1.29)

then the right side of Equation (1.28) can be rewritten to the form eiθ =

∞ X

k=0

(−1)k

∞

X θ2k θ2k+1 +i (−1)k . (2k)! (2k + 1)! k=0

(1.30)

TRIGONOMETRIC & HYPERBOLIC SINE & COSINE FUNCTIONS 7 Comparing Equations (1.14) and (1.30) gives the following expressions for the Taylor series of the cosine and sine functions cos θ =

∞ X

(−1)k

k=0

θ2k (2k)!

θ2 θ4 θ6 + − + ··· , 2! 4! 6! ∞ X θ2k+1 sin θ = (−1)k (2k + 1)! =1−

(1.31)

k=0

=θ−

1.3.4

θ5 θ7 θ3 + − + ··· . 3! 5! 7!

(1.32)

Addition and Subtraction Rules

We now derive formulas for the cosine and sine functions corresponding to the sum and difference of two angles. This can be easily done by starting with the relation (1.33) (eiθ1 )(eiθ2 ) = ei(θ1 +θ2 ) . Expanding the terms gives (eiθ1 )(eiθ2 ) = (cos θ1 + i sin θ1 )(cos θ2 + i sin θ2 ) = (cos θ1 cos θ2 − sin θ1 sin θ2 ) + i(cos θ1 cos θ2 + sin θ1 cos θ2 ),

(1.34)

and ei(θ1 +θ2 ) = cos(θ1 + θ2 ) + i sin(θ1 + θ2 ).

(1.35)

Comparison of the real and imaginary parts of Equations (1.34) and (1.35) gives cos(θ1 + θ2 ) = cos θ1 cos θ2 − sin θ1 sin θ2 , sin(θ1 + θ2 ) = sin θ1 cos θ2 + cos θ1 sin θ2 ,

(1.36a) (1.36b)

and these are called the addition relations for the sine and cosine functions. Changing the sign of θ2 , i.e., θ2 → −θ2 , gives the subtraction relations cos(θ1 − θ2 ) = cos θ1 cos θ2 + sin θ1 sin θ2 , sin(θ1 − θ2 ) = sin θ1 cos θ2 − cos θ1 sin θ2 .

1.3.5

(1.37a) (1.37b)

Product Rules

Adding Equations (1.36a) and (1.37a) gives   1 cos θ1 cos θ2 = [cos(θ1 + θ2 ) + cos(θ1 − θ2 )]. 2

(1.38)

8 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS Likewise, adding Equations (1.36b) and (1.37b) gives   1 sin θ1 cos θ2 = [sin(θ1 + θ2 ) + sin(θ1 − θ2 )]. 2 Similarly, subtracting Equation (1.36a) from Equation (1.37a) gives   1 [cos(θ1 − θ2 ) − cos(θ1 + θ2 )]. sin θ1 sin θ2 = 2

(1.39)

(1.40)

The formulas given in Equations (1.38), (1.39), and (1.40) are the product relations for the cosine and sine functions.

1.4

SINE AND COSINE: DYNAMIC SYSTEM APPROACH

A third way of investigating the cosine and sine functions is to consider them to be solutions to a dynamic system. For this situation, the dynamic system turns out to be the undamped harmonic oscillator studied in every elementary physics course. It is important to note that the independent variable is now the “time,” rather than the angle θ. To begin, let Equation (1.2) be rewritten to the form   1 1 (x2 + y 2 ) = , (1.41) H(x, y) = 2 2 where H(x, y) is the Hamiltonian of the system. This particular functional structure is a rescaled version of the harmonic oscillator energy function     y¯2 1 1 2 ¯ H(¯ x, y¯) = k¯ x = kA2 , (1.42) + 2m 2 2 where x ¯ = x¯(t¯) and y¯ = y¯(t¯), the initial conditions are x ¯(0) = A and y¯(0) = 0, and m is the mass of the object subject to a linear spring characterized by a spring constant k. The variable x¯(t¯) is the location of the mass m at time t¯, and y¯(t¯) = md¯ x(t¯)/dt¯ is the momentum. Returning to Equation (1.41), the dynamics is determined by the following pair of first-order differential equations ∂H(x, y) dx = , dt ∂y

dy ∂H(x, y) =− , dt ∂x

(1.43)

dy = −x. dt

(1.44)

which upon evaluation gives dx = y, dt

TRIGONOMETRIC & HYPERBOLIC SINE & COSINE FUNCTIONS 9 These two differential equations are equivalent to the following single, secondorder equation d2 x + x = 0, (1.45) dt2 for which the general solution is x(t) = C1 cos t + C2 sin t.

(1.46)

Using the initial conditions x(0) = A = 1,

y(0) = 0,

(1.47)

to evaluate the integration constants, gives C1 = 1,

C2 = 0

(1.48)

and x(t) = cos t,

y(t) = − sin t.

(1.49)

We thus see that the cosine and sine functions appear as solutions to the harmonic oscillator dynamic system.

1.4.1

x-y Phase-Space

Let us now examine in more detail the phase-space for the harmonic oscillator. This phase-space is the set of all trajectories in the x-y plane. These curves satisfy a first-order differential equation and we now provide two related ways to derive this equation. First, note that the curves are given by y = φ(x) and taking its time derivative gives dφ dx dy dx dy = = , (1.50) dt dx dt dx dt which can be solved for dy/dx to obtain the result dy/dt dy = . dx dx/dt

(1.51)

From Equation (1.44), it follows that dy =− dx

  x . y

(1.52)

A second way to derive Equation (1.52) is to take the derivative of the Hamilton, Equation (1.41), with respect to x, where y = y(x), to obtain d dy H(x, y(x)) = x + y = 0, dx dx which upon solving for dy/dx gives Equation (1.52).

(1.53)

10 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS The fixed point, (x∗ , y ∗ ), are constant solutions and they can be calculated from the conditions dx dy = 0, = 0. (1.54) dt dt Referring to Equation (1.44), we see there is only one fixed point, i.e., x∗ (t) = 0,

y ∗ (t) = 0.

(1.55)

For the harmonic oscillator, the fixed point represents the rest state of the oscillator.

1.4.2

Symmetry Properties of Trajectories in Phase-Space

Inspection of either the Hamiltonian, Equation (1.41), or the differential equation determining the trajectories in phase-space, Equation (1.52), shows that both are invariant under the following transformations T1 : x → −x,

y → y,

T2 : x → x, y → −y, T3 = T1 T2 : x → −x, y → −y.

(1.56) (1.57) (1.58)

These transformations have the following interpretations (see Figure 1.4.1): (i) T1 corresponds to reflection of a point P through the y-axis. (ii) T2 is reflection through the x-axis. (iii) T3 = T1 T2 is reflection through the origin. Figure 1.4.2 shows the results of these transformations on segments of curves in phase-space.

1.4.3

Null-Clines

For the Hamiltonian H(x, y), the equations of motion are dx ∂H(x, y) = ≡ f (x, y), dt ∂y dy ∂H(x, y) =− ≡ g(x, y). dt ∂x Thus, the differential equation for the trajectories in phase-space is dy dy/dt g(x, y) = = . dx dx/dt f (x, y)

(1.59) (1.60)

(1.61)

The null-clines are curves in phase-space where the derivative, dy/dx, is either zero or unbounded (infinite). They are determined by the requirements dy = 0 : g(x, y0 (x)) = 0, dx dy = ∞ : f (x, y∞ (x)) = 0. dx

(1.62a) (1.62b)

TRIGONOMETRIC & HYPERBOLIC SINE & COSINE FUNCTIONS 11

FIGURE 1.4.1: Symmetry transformations, in the x-y phase-plane.

12 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS

FIGURE 1.4.2: Symmetry transformations for segments of curves.

TRIGONOMETRIC & HYPERBOLIC SINE & COSINE FUNCTIONS 13 Applying these considerations to the harmonic oscillator, we have   1 dx dy H(x, y) = (x2 + y 2 ) −→ = y, = −x 2 dt dt dy x =− , dx y dy = 0 : x = 0 or along the y-axis, dx dy = ∞ : x = 0 or along the x-axis. dx

1.4.4

(1.63a) (1.63b) (1.63c) (1.63d)

Geometric Proof that All Trajectories Are Closed

We now demonstrate that just using knowledge of the major features of the trajectories in phase-space, it follows from the differential equation dy/dx = (−x/y) that all the trajectories for the harmonic oscillator are simply closed curves. The major impact of this conclusion is that this implies that all the solutions are periodic. Consider the top, left diagram in Figure 1.4.4, where the point P is selected to be on the positive y-axis. The trajectory, passing through P , in the firstquadrant, must start with zero slope on the y-axis and then have a negative slope in the remainder of this quadrant. When it makes contact with the xaxis, it does so with a vertical, negative slope. Applying the transformation T2 , reflection in the x-axis, gives the third diagram. Note that the point P ′ is the reflection of P and thus lies below the origin the same distance as P lies above the origin. If the transform T1 , reflection in the y-axis, is applied to the third diagram, then we obtain the result of the fourth diagram, i.e., a simple closed curve. This result is very important because it implies that x(t) and y(t) are periodic solutions, with the possibility that the period may depend on the initial conditions. The discussions of this section are very general and are applicable to other dynamic systems. We will use them in other investigations to be given in this book.

14 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS

(+)

(- )

(- )

(+)

FIGURE 1.4.3: Null-clines and signs of dy/dx in the four quadrants for the harmonic oscillator dynamic system.

1.5

HYPERBOLIC SINE AND COSINE: DERIVED FROM SINE AND COSINE

Using the Euler formula and its complex conjugate, see Equation (1.14), the cosine and sine have the representations eiθ + e−iθ , 2 eiθ − e−iθ sin θ = , 2i

cos θ =

The functions obtained by replacing θ by iθ, where i = the so-called hyperbolic functions, defined as

(1.64a) (1.64b) √ −1, are related to

eθ + e−θ = cosh θ, 2 eθ − e−θ hyperbolic sine ≡ = sinh θ. 2

hyperbolic cosine ≡

(1.65a) (1.65b)

The exact relationship between the cosine/sine and the cosh/sinh is given by the relations cosh(θ) = cos(iθ), i sinh(θ) = sin(iθ). (1.66)

TRIGONOMETRIC & HYPERBOLIC SINE & COSINE FUNCTIONS 15

T1

FIGURE 1.4.4: Geometric proof of simple, closed phase-space trajectories for the harmonic oscillator dynamic system.

16 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS 10

8

6

4

2

-3

-2

-1

0

1

2

3

10

5

-3

-2

-1

1

2

3

-5

- 10

FIGURE 1.5.1: The hyperbolic cosine and sine functions. Note that cosh θ and sinh θ are, respectively, even and odd. Further, unlike the cosine and sine, they are not periodic and become unbounded as θ → (±)∞. See Figure 1.5.1. The cosh/sinh satisfy many relations which are similar to those for the cosine and sine function; see the Appendix for a listing of the more important formulas. However, one of the most important of these will now be derived. Observe that (cosh θ)2 and (sinh θ)2 are (cosh θ)2 = 2

(sinh θ) =

 

eθ + e−θ 2 eθ − e−θ 2

2 2

=

e2θ + e−2θ + 2 , 4

(1.67a)

=

e2θ + e−2θ − 2 , 4

(1.67b)

and, if we subtract the second expression from the first, then the following relation results (cosh θ)2 − (sinh θ)2 = 1, (1.68)

TRIGONOMETRIC & HYPERBOLIC SINE & COSINE FUNCTIONS 17 which is to be contrasted with (cos θ)2 + (sin θ)2 = 1 for the cosine and sine functions. Returning to Equation (1.67), we have   1 e2θ + e−2θ 1 (cosh)2 = + = [1 + cosh(2θ)], (1.69a) 2 4 2     1 e2θ + e−2θ 1 2 (sinh) = − + = [cosh(2θ) − 1], (1.69b) 2 4 2 and the interesting relation (cosh θ)2 + (sinh θ)2 = cosh(2θ).

(1.69c)

All of the hyperbolic formulae, given in the Appendix, can be derived using similar techniques.

1.6

HYPERBOLIC FUNCTIONS: DYNAMIC SYSTEM DERIVATION

Consider a system having the Hamiltonian   1 1 H(x, y) = (y 2 − x2 ) = . 2 2

(1.70)

The equations of motion are dx ∂H(x, y) = = y, dt ∂y

dy ∂H(x, y) =− = x, dt ∂x

(1.71)

with the initial conditions x(0) = 0, y(0) = 1,

(1.72a)

x(0) = 0, y(0) = −1.

(1.72b)

or

(For some insight into these particular selections, see Figure 1.6.1.) Therefore dx = y, dt

dy = x, dt

(1.73)

and both x and y satisfy the second-order differential equation d2 w = w, dt2

(1.74)

18 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS

3 2 1

-3

-2

-1

1

2

3

-1 -2 -3

FIGURE 1.6.1: Plot of y 2 − x2 = 1. whose general solution is w(t) = Ae+t + Be−t .

(1.75)

For the initial conditions given in Equation (1.72), we have the solutions x(0) = 0, y(0) = 1 : x(t) = sinh(t), y(t) = cosh(t), x(0) = 0, y(0) = −1 : x(t) = sinh(t), y(t) = − cosh(t).

1.7

(1.76a) (1.76b)

θ-PERIODIC HYPERBOLIC FUNCTIONS

We now consider the nonstandard case of constructing periodic hyperbolictype solutions for the equation y 2 − x2 = 1.

(1.77)

Figure 1.7.1 defines the variables r and θ associated with a point (x, y) located on the curve represented by Equation (1.77).

TRIGONOMETRIC & HYPERBOLIC SINE & COSINE FUNCTIONS 19 3 2

P(x,y)

1

-3

-2

-1

1

2

3

-1 -2 -3

FIGURE 1.7.1: Definition of the variables r and θ. We define three periodic hyperbolic functions as follows: pch(θ) ≡ x,

psh(θ) ≡ y, pdh(θ) ≡ r.

(1.78a) (1.78b) (1.78c)

The expressions on the left side are, respectively, called the periodic hyperbolic cosine, the periodic hyperbolic sine, and the periodic hyperbolic dine functions. There are two functional relations connecting x, y and r; they are ( r2 = x2 + y 2 (1.79) y 2 − x2 = 1. Note, we can also express x and y as x(θ) = r(θ) cos θ,

y(θ) = r(θ) sin θ,

(1.80)

where r(θ) can be determined by substituting these representations into the second of Equation (1.79), i.e., r(θ)2 [(sin θ)2 − (cos θ)2 ] = 1. Therefore,

1 r(θ) = p , 2 (sin θ) − (cos θ)2

(1.81)

(1.82)

20 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS but, restrictions must be placed on this result. Since r(θ) is the distance from the origin to a point on the curve y 2 − x2 = 1, the following angular intervals are not allowed  π  0 < θ < 4 , 3π 5π (1.83) 4 < θ < 4 ,   7π 4 < θ < 2π. The reason for this is that for these angular intervals, there is no curve to be intercepted by a ray from the origin and r(θ) is not defined. An alternative way to obtain the non-allowed angular regions is to examine the expression in the radical for Equation (1.82), i.e.,     1 1 (sin θ)2 − (cos θ)2 = (1 − cos 2θ) − (1 + cos 2θ) 2 2 = − cos 2θ

(1.84)

The requirement that − cos 2θ > 0,

(1.85)

gives the results listed in Equation (1.83). The explicit analytical expressions can now be given for the periodic hyperbolic functions; they are 1 r(θ) = pdnh(θ) = p , 2 (sin θ) − (cos θ)2 cos θ x(θ) = pch(θ) = p , (sin θ)2 − (cos θ)2 sin θ y(θ) = psh(θ) = p , (sin θ)2 − (cos θ)2

(1.86a) (1.86b) (1.86c)

where the angular intervals for which they are defined are listed in Equation (1.83). Finally, it can be easily seen that the three functions become unbounded at the boundaries of the angular intervals where they are defined; see Figure 1.7.2, where r(θ), x(θ), and y(θ) are shown over the interval, 0 ≤ θ ≤ 2π.

1.8

DISCUSSION

Let us examine what we have found in the work given in this chapter. Of importance is the fact that the equation x2 + y 2 = 1,

(1.87)

TRIGONOMETRIC & HYPERBOLIC SINE & COSINE FUNCTIONS 21

5

x( ) 0 -1.8 0

/4

/2

3 /4

5 /4

3 /2

7 /4

0

/4

/2

3 /4

5 /4

3 /2

7 /4

0

/4

/2

3 /4

5 /4

3 /2

7 /4

2 1.5

y( ) 1 0.5 0

2

r( )

1 0

FIGURE 1.7.2: Plots of r(θ) and x(θ) from Equations (1.86).

22 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS can be interpreted in several different ways. First, from the perspective of plane geometry, this equation represents a circle. Second, this equation can be considered (proportional to) the Hamiltonian for the harmonic oscillator dynamic system, which gives a model for a broad range of phenomena in the natural sciences. Third, Equation (1.87) can be taken as a functional equation and, with this interpretation, it allows for the construction of many different classes of solutions, periodic and non-periodic, and, continuous and non-continuous. Chapter 5 gives a discussion of certain classes of periodic solutions. Similar results hold for the equation y 2 − x2 = 1,

(1.88)

for which unbounded periodic and non-periodic solutions exist. In the chapters to come, we apply the general methodology of this chapter to investigate solutions to |y|p ± |x|q = 1, (1.89) where p > 0,

q > 0.

(1.90)

NOTES AND REFERENCES Sections 1.2, 1.3, and 1.5: The elementary properties of the trigonometric cosine and sine, and hyperbolic cosine and sine functions can be found in any standard textbook covering elementary geometry and trigonometry. Section 1.4: A brief introduction to Hamiltonian dynamics is given in the Appendix, Section E. Sections 1.6 and 1.7: As far as I am aware, the results given in these sections is new, and generalize the discussions in Sections 1.2, 1.3, and 1.4.

Chapter 2 ELLIPTIC FUNCTIONS

2.1

INTRODUCTION

The main purpose of this chapter is to introduce the standard Jacobi elliptic functions and show that other related elliptic functions also exist. Further, we demonstrate that some such functions are periodic, while others are non-periodic. Section 2.2 examines the construction of what we name θ-periodic elliptic functions. These functions are based on the geometric properties of the ellipse y2 +

x2 = 1, a2

a > 1.

(2.1)

In Section 2.3, we investigate the case where the expression in Equation (2.1) is taken to be the Hamiltonian of a one-space-dimensional dynamic system. The standard Jacobi elliptic functions are investigated in some detail in Section 2.4. We also provide a listing of many of their important mathematical properties in Section 2.5. Section 2.6 presents a discussion of the Jacobi elliptic functions as the solutions to a three-dimensional dynamic system, for which the equations of motion are three coupled, nonlinear, first-order, ordinary differential equations. The next two Sections, 2.7 and 2.8, consider “hyperbolic” generalizations of the work given in Sections 2.2 and 2.3. Finally, in Section 2.9, we give a general summary of what was done and its significance.

2.2

θ-PERIODIC ELLIPTIC FUNCTIONS

Consider the graph of the elliptic curve given in Figure 2.2.1. Let P (x, y) be a point on this curve and let r denote a ray from the origin to P (x, y), where θ is the angle between the ray and the positive x-axis. The following 23

24 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS

(0,1)

2

FIGURE 2.2.1: Graph of y 2 + xa2 = 1, and the definitions of r and θ. P (x, y) denotes a point on the ellipse, and a > 1. two relations hold x2 = 1, a2 r2 = x2 + y 2 ,

(2.2b)

x(θ) = r(θ) cos θ,

(2.3a)

y(θ) = r(θ) sin θ, d(θ) = r(θ).

(2.3b) (2.3c)

y2 +

(2.2a)

where x(θ), y(θ), and d(θ) are

Substituting Equations (2.3a) and (2.3b) into Equation (2.2a) gives  r2 1 = r sin(θ) + (cos θ)2 a2  2   r = r2 1 − (cos θ)2 + (cos θ)2 a2   = r2 1 − k 2 (cos θ)2 , 2

2



(2.4)

ELLIPTIC FUNCTIONS where k 2 is defined to be k2 ≡ 1 −

1 . a2

25

(2.5)

Since a > 1, we have the bounds 0 < k 2 < 1,

a > 1.

(2.6)

Note that a = 1 reduces the ellipse to a circle. Solving Equation (2.4) for r(θ) gives

and it follows that

1 r(θ) = p , 1 − k 2 (cos θ)2 1 ≤ r(θ) ≤ a,

0 ≤ θ ≤ 2π,

(2.7)

(2.8)

where we have suppressed, for the time being, the dependence on k 2 . The elliptic dine, cosine, and sine periodic functions are defined by the expressions 1 , edn(θ, k 2 ) ≡ r(θ) = p 1 − k 2 (cos θ)2 cos θ ecn(θ, k 2 ) ≡ r(θ) cos θ = p , 1 − k 2 (cos θ)2 sin θ esn(θ, k 2 ) ≡ r(θ) sin θ = p . 1 − k 2 (cos θ)2

These three functions have the properties: 1) bounds  2  1 ≤ edn(θ, k ) ≤ a −a ≤ ecn(θ, k 2 ) ≤ a   −1 ≤ esn(θ, k 2 ) ≤ 1

(2.9) (2.10) (2.11)

(2.12)

2) even/odd-ness

3) periodicity

 2 2  edn(−θ, k ) = edn(θ, k ) ecn(−θ, k 2 ) = ecn(θ, k 2 )   esn(−θ, k 2 ) = (−)esn(θ, k 2 )  2 2  edn(θ + π, k ) = edn(θ, k ) ecn(θ + 2π, k 2 ) = ecn(θ, k 2 )   esn(θ + 2π, k 2 ) = esn(θ, k 2 )

(2.13)

(2.14)

26 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS 4) special values (
edn(0, k 2 ) = a, edn π2 , k 2 = 1, edn(π, k 2 ) = a
2 edn 3π = 1, edn(2π, k 2 ) = a 2 ,k (
ecn(0, k 2 ) = a, ecn π2 , k 2 = 0, ecn(π, k 2 ) = −a
2 ecn 3π = 0, ecn(2π, k 2 ) = a 2 ,k (
esn(0, k 2 ) = 0, esn π2 , k 2 = 1, esn(π, k 2 ) = 0
2 esn 3π = −1, esn(2π, k 2 ) = 0. 2 ,k

(2.15a) (2.15b) (2.15c)

Figure 2.2.2 provides graphs of these three functions.

2.3

ELLIPTIC HAMILTONIAN DYNAMICS

Consider a dynamic system having the Hamiltonian    1 x2 1 2 H(x, y) = y + 2 = . 2 a 2

(2.16)

(Compare this to the expression in Equation (2.1).) The equations of motion are dx ∂H(x, y) = = y, dt ∂y   dy ∂H(x, y) 1 =− =− x, dt ∂x a2

(2.17a) (2.17b)

and the initial values are selected to be x(0) = a,

y(0) = 0.

(2.18)

Equations (2.17) implies that x(t) satisfies the second-order, linear, differential equation   d2 x 1 + x = 0, (2.19) 2 dt a2 with initial conditions

dx(0) = 0. dt

(2.20)

    t t + c2 sin . a a

(2.21)

x(0) = a, Therefore, x(t) is x(t) = c1 cos

ELLIPTIC FUNCTIONS

27

2.0 1.5 1.0 0.5 0.0

π 2

π

3π 2

2π

π 2

π

3π 2

2π

2

1

0

-1

-2

2

1

0

π 2

π

3π 2

2π

-1

-2

FIGURE 2.2.2: Graphs of edn(θ), ecn(θ), and esn(θ) for a =

3 2

or k 2 = 59 .

28 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS and imposing the initial conditions gives the periodic solutions     t t x(t) = a cos , y(t) = − sin . a a

(2.22)

Physically, x(t) and y(t) correspond to the position and velocity of a linear harmonic oscillator having a period T = 2πa.

2.4

(2.23)

JACOBI, CN, SN, AND DN FUNCTIONS

We will now construct and examine the properties of the so-called Jacobi elliptic functions. Refer to Figure 2.2.1 for the definitions of r, θ, and a > 1. The eccentricity of the ellipse is defined to be r 1 ǫ ≡ k = 1 − 2 , a > 1. (2.24) a Note that for a = 1 the ellipse becomes a circle. The three Jacobi elliptic functions are defined in the following way, in terms of (x, y, r, a): cn ≡

x , a

sn ≡ y,

dn ≡

r > 0. a

(2.25)

Examination of Figure 2.2.1 shows that all three of these functions are periodic in θ, with period 2π. However, this is not the independent variable selected for these functions. The standard independent variable, u, is selected to be Z θ u(θ) = r(ψ)dψ, (2.26) 0

where r(θ) is the length of a ray from the origin to the ellipse, where θ is the angle that the ray makes with the positive x-axis. While not explicitly indicated, u depends also on k 2 , as defined by Equation (2.5). This particular definition of u(θ) is based on construction of the Jacobi elliptic functions in terms of integral representations. See the note to this section for pertinent references on this issue. One consequence of this definition is that when ǫ = k = 0, i.e., a = r = 1, u(θ) is Z θ u(θ) = (1)dψ = θ, (2.27) 0

and

cn = cos θ,

sn = sin θ,

dn = 1.

(2.28)

ELLIPTIC FUNCTIONS

29

Thus, as expected, the elliptic functions reduce to trigonometric functions and the above definition of u(θ) is consistent with the a priori known geometrical features of the problem. From the fact that r(−θ) = r(θ), (2.29) it is an easy calculation to show that u(−θ) = −u(θ).

(2.30)

Using the results in Equations (2.25), (2.26), and (2.30) allows the following results to be stated: (i) there are three elliptic functions and they are denoted (cn, sn, dn); (ii) these three elliptic functions depend on the independent variable, u, and the parameter k, i.e., cn = cn(u, k),

sn = sn(u, k),

dn = dn(u, k);

(iii) the parameter k, called the modulus or the eccentricity, ǫ, satisfies the bounds, 0 ≤ k ≤ 1; (iv) when a → 1, or k = 0, the elliptic functions reduce to the trigonometric functions; (v) the three elliptic functions are periodic in θ, with period 2π; however, they are periodic in u with period T (which for the time being is unknown, but will be calculated later). The three elliptic functions, as we have constructed them, are called the Jacobi, “cosine,” “sine,” and “dine” functions.

2.4.1

Elementary Properties of Jacobi Elliptic Functions

From the definitions of the Jacobi elliptic functions, see Equations (2.25), (2.29), and (2.30), we conclude that cn and dn are even functions of u, while sn is an odd function of u, i.e., cn(−u) = cn(u),

sn(−u) = −sn(u),

dn(−u) = dn(u).

(2.31)

Using the definitions given by Equation (2.25) in the relations presented in Equations (2.2) gives sn(u)2 + cn(u)2 = 1, 2

2

(2.32a)

2

dn(u) + k sn(u) = 1.

(2.32b)

The Jacobi elliptic functions satisfy the bounds −1 ≤ cn(u) ≤ 1,

−1 ≤ sn(u) ≤ 1,

1 ≤ dn(u) ≤ 1. a

(2.33)

30 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS

2.4.2

First Derivatives

From Figure 2.2.1, we have θ = tan−1 and from this it follows that dθ =



1 r2



y  x

,

(x dy − y dx),

and from taking the derivative of Equation (2.27)   1 du = r dθ = (x dy − y dx). 4

(2.34)

(2.35)

(2.36)

Likewise, from Equation (2.1) y dy +

x dx = 0, a2

and from this, we obtain    2  x a y dy = − dx or dx = − dy. a2 y x

(2.37)

(2.38)

Substituting, respectively, the results in Equations (2.38) into (2.36), and simplifying gives " # dx ry =− = −ry, (2.39)
x 2 du + y2 a

dy rx = 2 du x + a2 y 2 " #  rx  1 =
x 2 a2 + y2 a rx = 2 a

or

d cn = −dn sn, du

d sn = dn cn. du

(2.40)

(2.41)

Now d 2 d 2 (r ) = (x + y 2 ) du du dr dx dy r =x +y du du du

(2.42)

ELLIPTIC FUNCTIONS

31

and

 x  dx  y  dy dr = + . (2.43) du r du r du Using the definitions from Equations (2.25), this equation can be written as d dn = −k 2 cn sn. du

(2.44)

Observe that if we take the limit a → 1 or k 2 → 0, then these expressions for the derivatives become d cos θ = − sin θ, dθ

d sin θ = cos θ, dθ

dr(θ) = 0, dθ

(2.45)

i.e., they reduce to the standard results for the usual trigonometric functions.

2.4.3

Differential Equations

All three Jacobi elliptic functions satisfy nonlinear, second-order, ordinary differential equations. We now derive these equations for (sn, cn, dn), where it is understood that sn = sn(u, k 2 ), etc. Using Equations (2.32a),b, we have d sn = cn dn du  p  p 1 − sn2 1 − k 2 sn2 , =

(2.46)

and replacing sn(u) by y(u), this equation becomes 

dy du

2

= (1 − y 2 )(1 − k 2 y).

(2.47)

Taking the derivative and simplifying the resulting expression gives d2 y + (1 + k 2 )y − 2k 2 y 3 = 0, du2

(2.48)

which is the nonlinear, second-order, differential equation for which with the initial values dy(0) = dn(0)cn(0) = 1, (2.49) y(0) = 0, du gives the solution y(u) = sn(u). Let x(u) = a cn(u), then substituting this into Equation (2.41) and squaring, gives  2   dx (2.50) = (1 − k 2 ) + k 2 x2 (1 − x2 ). du

32 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS Taking the derivative of the expression, canceling common factors, and simplifying gives the following nonlinear, second-order differential equation for x(u) d2 x + (1 − 2k 2 )x + 2k 2 x3 = 0, (2.51) du2 for the initial conditions x(0) = a,

dx(0) = 0. du

(2.52)

A similar calculation for w(u) = dn(u) gives d2 w − (2 − k 2 )w + 2w3 = 0, du2

(2.53)

with

dw(0) = 0. (2.54) du In summary, if V (u) is any of the three Jacobi elliptic functions, (cn, sn, dn), then V (u) satisfies first- and second-order derivative relations which take the form  2 dV = A + BV 2 + CV 4 , (2.55a) du d2 V + αV + βV 3 = 0, (2.55b) du2 w(0) = 1,

where (A, B, C) are functions of k 2 , and (α, β) are determined by (A, B, C), and therefore are functions of k 2 .

2.4.4

Calculation of u(θ) and the Period for cn, sn, dn

The calculations and analysis given above makes it clear that u, as defined by Equation (2.26), is the appropriate independent variable for the Jacobi elliptic functions. We now calculate u(θ). From Figure 2.2.1, we have x = r cos θ and y = r sin θ. Therefore from r2 = x2 + y 2 , it follows that

and therefore

y2 +

x2 = 1, a2

a r(θ) = p , 2 1 + (a − 1)(sin θ)2 u(θ) = a

Z

0

θ

dψ p . 1 + (a2 − 1)(sin ψ)2

(2.56)

(2.57)

(2.58)

Using the “look-up” method, i.e., seeing if the integral appears in one of the

ELLIPTIC FUNCTIONS

33

standard reference books on mathematical tables and formulas, we find this integral in Gradshteyn and Ryzhik (see page 173, formula 2.597.1), and obtain the result       a sin θ ,k ,   (2.59) u(θ) = F sin−1  2   (sin θ)2 1+ k 1−k2

where F (φ, k) is an elliptic integral of the first kind. Remember that a and k are related; see Equation (2.24). If θ = π/2, then π  π  u =F , k = K(k), (2.60) 2 2 where K(k) is the complete elliptic integral of the first kind. Both F (φ, k) and K(k) are tabulated for values of φ and k, respectively, in the intervals, 0 ≤ φ ≤ π/2 and 0 ≤ k ≤ 1. Since, in terms of the variable θ, the period of the Jacobi elliptic functions is 2π, and the corresponding period in the variable u is π  π  = 4F , k = 4K(k). (2.61) T = 4u 2 2 Thus, if ef (u) denotes any of the three Jacobi elliptic functions, (cn, sn, dn), then the following periodicity condition holds ef (u + 4K(k)) = ef (u).

2.4.5

(2.62)

Special Values of Jacobi Elliptic Functions

In addition to the upper and lower bounds, expressed in Equation (2.33), these functions are zero or take their respective maximum/minimum magnitudes at the following indicated multiples of K = K(k): sn(0) = 0 sn(K) = 1 sn(2K) = 0 sn(3K) = −1 sn(4K) = 0

cn(0) = 1 cn(K) = 0 cn(2K) = −1 cn(3K) = 0 cn(4K) = 1

dn(0) = 1 dn(K) = a1 dn(2K) = 1 dn(3K) = a1 dn(4K) = 1

Also, note in summary: (i) When k = 0, then K(0) = π/2, and the three Jacobi elliptic functions reduce to (cos θ, sin θ, 1). (ii) The fundamental period of cn and sn is 4K(k), while that of dn is 2K(k). (iii) sn has zeros at even multiples of K, while cn has zeroes at odd multiples of K. (iv) sn has maximum magnitudes at odd multiples of K, while cn has its maximum magnitude at even multiples of K. For each, the maximum magnitudes are one.

34 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS (v) dn is positive for k values in the interval, 0 ≤ k < 1. (vi) Since 1 ≤ r(θ) ≤ a, then du = r(θ) > 0, dθ θ ≤ u(θ) ≤ aθ,

(2.63a) a > 1.

(2.63b)

Consequently, u(θ) is a monotonic increasing function of θ, bounded by the two functions u1 (θ) = θ and u2 (θ) = aθ. See Figure 2.4.1 for graphs of sn(u), cn(u), and dn(u).

2.5 2.5.1

ADDITIONAL PROPERTIES OF JACOBI ELLIPTIC FUNCTIONS Fundamental Relations for Square of Functions cn2 + sn2 = 1 2

2

2

cn + (1 − k )sn = 1 dn2 + k 2 sn2 = 1

2.5.2

(2.64) (2.65) (2.66)

Addition Theorems cn(x)cn(y) − sn(x)sn(y)dn(x)dn(y) 1 − k 2 sn2 (x)sn2 (y) sn(x)cn(y)dn(y) + sn(y)cn(x)dn(x) sn(x + y) = 1 − k 2 sn2 (x)sn2 (y) dn(x)dn(y) − k 2 sn(x)sn(y)cn(x)cn(y) dn(x + y) = 1 − k 2 sn2 (x)sn2 (y) cn(x + y) =

(2.67) (2.68) (2.69)

ELLIPTIC FUNCTIONS

35

1.0 0.5 -4

-2

- 0.5

2

4

2

4

2

4

- 1.0

1.0 0.5 -4

-2

- 0.5 - 1.0

1.0 0.5 -4

-2

- 0.5 - 1.0

FIGURE 2.4.1: Graphs of the three Jacobi elliptic functions for k = 0.8.

36 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS

2.5.3

Product Relations

sn(u + v) · sn(u − v) = cn(u + v) · cn(u − v) = dn(u + v) · dn(u − v) = sn(u ± v) · cn(u ∓ v) = sn(u ± v) · dn(u ∓ v) = cn(u ± v) · dn(u ± v) =

2.5.4

2.5.5

sn2 u − sn2 v 1 − k 2 sn2 u sn2 v cn2 u − sn2 v dn2 u 1 − k 2 sn2 u sn2 v dn2 u − k 2 cn2 u sn2 v 1 − k 2 sn2 u sn2 v sn u cn u dn v ± sn v cn v dn u 1 − k 2 sn2 u sn2 v sn u dn u cn v ± sn v dn v cn u 1 − k 2 sn2 u sn2 v cn u dn u cn v ∓ k ′2 sn u sn v 1 − k 2 sn2 u sn2 v

(2.70) (2.71) (2.72) (2.73) (2.74) (2.75)

cn, sn, dn for Special k Values

sn(u, 0) = sin u cn(u, 0) = cos u

sn(u, 1) = tanh u cn(u, 1) = sech u

(2.76) (2.77)

dn(u, 0) = 1

dn(u, 1) = sech u

(2.78)

Fourier Series

Let

(

q = exp v=

πu 2K ,



−πk′ K



,

(2.79)

where K = K(k 2 ) and K ′ = K(1 − k 2 ), then cn(u), sn(u), and dn(u) have the expansions ! X  1 ∞ q n+ 2 2π cos[(2n + 1)v] (2.80) cn(u) = K · k n=0 1 + q 2n+1 ! X 1 ∞ 2π q n+ 2 sn(u) = sin[(2n + 1)v] K · k n=0 1 − q 2n+1   X   ∞  qn 2π 2π + cos(2nv) dn(u) = 2K K n=1 1 + q 2n   X   ∞  nq n 2π 2 E − K(k ′ )2 2 + cos(2nv) cn = K · k2 K 2 k 2 n=1 1 − q 2n 

(2.81)

(2.82)

(2.83)

ELLIPTIC FUNCTIONS    X  ∞  K −E nq n 2π 2 sn2 = − cos(2nv) K · k2 K 2 k 2 n=1 1 − q 2n dn2 =



E K



+



2π 2 K2

X ∞  n=1

nq n 1 − q 2n



cos(2nv).

37 (2.84)

(2.85)

In the above, K and E are complete elliptic integrals of the first and second kind.

2.6

DYNAMICAL SYSTEM INTERPRETATION OF ELLIPTIC JACOBI FUNCTIONS

An alternative derivation and interpretation of the three Jacobi elliptic functions, (cn, sn, dn), is to consider them as the solution to a threedimensional dynamical system. This means that they are represented as the solutions to three coupled, nonlinear, first-order differential equations. This methodology was summarized in a paper by Meyer who based his work on the earlier results of Hille and Chicone. This section provides a concise summary of Meyer’s presentation, and for the most part, his notation will be used. Anyone desiring the full proofs and other analytical background details can consult the following references: 1. E. Hille, Lectures on Ordinary Differential Equations (Addison-Wesley, Reading, MA, 1969). 2. C. Chicone, Ordinary Differential Equations with Applications (Dover, New York, 1961). 3. K. R. Meyer, Jacobi elliptic functions from a dynamical systems point of view, American Mathematical Monthly 108 (2001), 729–737.

2.6.1

Definition of the Dynamic System

Let k be a real number such that 0 < k < 1.

(2.86)

38 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS The dynamic system is defined by the three coupled, nonlinear differential equations dx = yz, dt dy = −zx, dt dz = −k 2 xy, dt where the initial conditions are taken to be x(0) = 0,

2.6.2

2.6.3

(2.87b) (2.87c)

z(0) = 1.

(2.88)

Limits k → 0+ and k → 1−

We have

and

y(0) = 1,

(2.87a)

    x(t) sin t Lim y(t) = cos t , k→0+ z(t) 1     tanh(t) x(t) Lim y(t) =  sech(t)  . k→1− sech(t) z(t)

(2.89)

(2.90)

First Integrals

The functions I(x, y) and J(x, y), defined as I(x, y) ≡ x2 + y 2 , 2 2

(2.91) 2

J(x, z) ≡ k x + z ,

(2.92)

are first integrals of Equations (2.87), i.e., let     x(t) φ1 (t) y(t) = φ2 (t) z(t) φ3 (t)

(2.93)

be any solution of Equation (2.87), then

I(φ1 (t), φ2 (t)) = constant, J(φ1 (t), φ3 (t)) = constant.

(2.94a) (2.94b)

d dx dy I(x(t), y(t)) = 2x + 2y dt dt dt = 2x(yz) + 2y(−zx) = 0,

(2.95)

Proof:

ELLIPTIC FUNCTIONS

39

and dx dz d J(x(t), z(t)) = 2k 2 x + 2z dt dt dt = 2k 2 x(yz) + 2z(−k 2 xy) = 0.

2.6.4

(2.96)

Bounds and Symmetries

The following statements are true: (i) The functions, (x(t), y(t), z(t)), are periodic in t. (ii) For fixed k, 0 < k < 1, then for all t > 0 ( x2 (t) + y 2 (t) = 1, k 2 x2 (t) + z 2 (t) = 1.

(2.97)

(iii) The following inequalities are satisfied −1 ≤ x(t) ≤ 1,

−1 ≤ y(t) ≤ 1,

(iv) If (x(t), y(t), z(t)) is a solution   (−x(t), (x(−t),   (x(−t),

p 1 − k 2 ≤ z(t) ≤ 1.

(2.98)

of Equations (2.87), then so are y(−t), z(−t)) −y(−t), z(−t)) y(−t), −z(−t)).

(2.99)

(v) For fixed k, 0 < k < 1, we have x(−t) = −x(t),

y(−t) = y(t),

z(−t) = z(t).

(vi) There exists a constant, K(k), such that   x(t + 4K) = x(t), y(t + 4K) = y(t),   z(t + 2K) = y(t),

(2.100)

(2.101)

i.e., x(t) and y(t) are periodic, with period 4K, while z(t) is periodic, with period 2K.

40 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS

2.6.5

Second-Order Differential Equations

The following nonlinear, second-order, differential equations are satisfied by (x(t), y(t), z(t)) ( 2 d x 2 2 3 dt2 + (1 + k )x − 2k x = 0, (2.102) x(0) = 0, dx(0) dt = 1, d2 y dt2

+ (1 − 2k 2 )y + 2k 2 y 3 = 0, y(0) = 1, dy(0) dt = 0,

(

− (2 − k 2 )z + 2z 3 = 0, z(0) = 1, dz(0) dt = 0.

(

2.6.6

d2 z dt2

(2.103)

(2.104)

Discussion

Comparison of the results derived by Meyer, as summarized in this section, with the results given in Section 2.4, shows that the correspondence between (sn, cn, dn) and (x, y, z) is sn ↔ x,

cn ↔ y,

dn ↔ z.

(2.105)

It is this correspondence which allows Meyer to state that the Jacobi elliptic functions can be interpreted as solutions to a three dimensional dynamic system. A major feature of Meyer’s article is that he demonstrates that the Jacobi elliptic functions can be applied to several important problems arising in mathematics, physics, and engineering; these include: • the pendulum system, • evaluation of elliptic integrals, and • physical systems having quadratic or cubic forces.

2.7

HYPERBOLIC ELLIPTIC FUNCTIONS AS A DYNAMIC SYSTEM

Consider a dynamic system modeled by the Hamiltonian    x 2  1 1 H(x, y) = y2 − = . 2 a 2

(2.106)

ELLIPTIC FUNCTIONS

41

The equations of motion are dx ∂H = = y, dt ∂y   dy ∂H 1 =− = x, dt ∂x a2

(2.107a) (2.107b)

with the initial conditions x(0) = 0,

y(0) = 1,

(2.108)

y(0) = −1.

(2.109)

or x(0) = 0,

First, observe that both x(t) and y(t) satisfy the second-order differential equation   d2 w 1 = w, (2.110) dt2 a2 for which the general solution is w(t) = Ae(w/a) + Be−(w/a) ,

(2.111)

where A and B are arbitrary integration constants. Second, from Equation (2.111), the solutions corresponding to the two sets of initial conditions can be calculated. These solutions are given by the expressions (
x(t) = a sinh at ,
x(0) = 0, y(0) = 1 : (2.112) y(t) = cosh at , and

x(0) = 0,

(
x(t) = −a sinh xa ,
y(0) = −1 : y(t) = − cosh at .

(2.113)

Figures 2.7.1 and 2.7.2 present the graphs of these two solutions. Note that these solutions are the corresponding elliptic hyperbolic functions for the case where Equation (2.106) is considered the Hamiltonian of a one (space)-dimensional system.

2.8

HYPERBOLIC θ-PERIODIC ELLIPTIC FUNCTIONS

Consider the curve y2 −

 x 2 a

= 1,

(2.114)

42 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS

4

2

-2

-1

1

2

-2

-4

4

3

2

1

-2

-1

0

1

FIGURE 2.7.1: Graphs of Equations (2.112).

2

ELLIPTIC FUNCTIONS

43

4

2

-2

-1

1

2

-2

-4

1

-2

-1

1

-1

-2

-3

-4

FIGURE 2.7.2: Graphs of Equations (2.113).

2

44 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS

2

1

-2

-1

1

2

-1

-2

FIGURE 2.8.1: Graphs of y 2 −


x 2 a

= 1, and definition of r, θ, and P (x, y).

which forms the basis for the treatment of the hyperbolic θ-periodic elliptic functions. The variable r is the segment of the ray from the origin (0, 0), to a point P (x, y) on the curve given by Equation (2.114). Using ( r2 = x2 + y 2 , (2.115) x = r cos θ, y = r sin 0, then r(θ) can be calculated and its value is r(θ) = q 1−

1 1+a2 a2

.


(2.116)

(cos θ)2

Since r(θ) is only meaningful for r(θ) real and positive, this implies that the relevant θ values are those for which   1 + a2 1− (cos θ)2 > 0 (2.117) a2 or

|cos θ| < If ∆(a) is taken to be ∆(a) = cos−1

r

a2 . 1 + a2

r

a2 1 + a2

(2.118)

!

,

(2.119)

ELLIPTIC FUNCTIONS then the allowed angles are those for which π  π  −∆ 0.

(3.27)

and, on this interval,

Therefore, to calculate the ck , we must evaluate the integral   Z π/4 8 cos(4kθ) ck = · dθ. π cos θ + sin θ 0

(3.28)

Using the trigonometric relation cos(θ1 − θ2 ) = cos θ1 cos θ2 + sin θ1 sin θ2 ,

(3.29)

54 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS and setting, θ1 = θ and θ2 = π/4, gives    π 1 cos θ − (cos θ + sin θ), (3.30) = √ 4 2 √ a result based on cos(π/4) = sin(π/4) = 1/ 2. Therefore, Equation (3.28) can be expressed as  √  Z π/4 4 2 cos(4kθ)
· dθ. (3.31) ck = π cos θ − π4 0 Making the linear transformation

θ→ψ=θ− gives

π , 4

 √ Z 0 cos[4k(ψ + π4 )] 4 2 · dψ. ck = π cos ψ −π/4

Also,

h  π i = cos(4kψ + kπ) = (−1)k cos(4kψ), cos 4k ψ + 4 and thus allows Equation (3.33) to now be written as  √ Z 0 4 2 cos(4kψ) k ck = (−1) · dψ. π cos ψ −π/4

(3.32)

(3.33)

(3.34)

(3.35)

If we replace ψ by (−φ), i.e., then

ψ = −φ,

(3.36)

 √  Z π/4 4 2 cos(4kφ) ck = (−1) · dφ. π cos φ 0

(3.37)

k

So, what can be done to evaluate the integral in the last equation? A way to proceed is to construct the Fourier series for (cos θ)−1 . We now give arguments for the following expression: 1 = 2[cos θ − cos 3θ + cos 5θ − cos 7θ + · · · + (−1)k cos(2k + 1)θ + · · · ] cos θ ∞ X =2 (−1)k cos(2k + 1)θ. (3.38) k=0

Proof:

1=1 = 1 + cos 2θ − cos 2θ − cos 4θ + cos 4θ + cos 6θ − cos 6θ + · · · = (1 + cos 2θ) − (cos 2θ + cos 4θ) − (cos 4θ + cos 6θ) + · · · + (−1)n [cos(2nθ) + cos 2(n + 1)θ] + · · · .

(3.39)

SQUARE FUNCTIONS

55

Now cos(2nθ) + cos 2(n + 1)θ = 2 cos θ cos(2n + 1)θ,

(3.40)

and this relationship allows Equation (3.39) to be written as 1=

∞ X

(−1)k (2 cos θ) cos(2k + 1)θ

k=0

= 2 cos θ

∞ X

(−1)k cos(2k + 1)θ.

(3.41)

k=0

Finally, dividing by cos θ gives the result expressed in Equation (3.38). Substituting Equation (3.38) into Equation (3.37) gives  √ X Z π/4 ∞ 8 2 ck = (−1)k (−1)m cos(4kθ) cos[(2m + 1)θ]dθ. (3.42) π 0 m=0 After a direct, but tedious calculation, we find that the integral has the value Z π/4 cos(4kθ) cos[(2m + 1)θ]dθ 0    h  π i 1 1 = sin (4k + 2m + 1) 2 4k + 2m + 1 4    h  π i 1 1 sin (4k − 2m − 1) . (3.43) + 2 4k − 2m − 1 4 Note that since (2m + 1) is odd, then the denominators (4k ± 2m ± 1) cannot be zero. Therefore, we have for the Fourier coefficients, ck , the result ( 
  √ X ∞ sin (4k + 2m + 1) π4 4 2 k ck = (−1) π (4k + 2m + 1) m=0 )
  sin (4k − 2m − 1) π4 + . (3.44) (4k − 2m − 1) This means that the three square-trigonometric functions have Fourier series representation (see Equations (3.22), (3.23), and (3.24)), which can be calculated from the following relations: sqd(θ) = r(θ) =

∞

c0 X + ck cos(kθ), 2

(3.45a)

k=1

sqc(θ) = r(θ) cos θ,

(3.45b)

sqs(θ) = r(θ) sin θ.

(3.45c)

We leave as an exercise, for the reader to complete, the explicit calculation of the Fourier coefficients for sqc(θ) and sqs(θ).

56 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS

3.5

DYNAMIC SYSTEM INTERPRETATION OF |x|+ |y| = 1

Consider a dynamic system modeled by the following Hamiltonian H(x, y) = |x| + |y| = 1.

(3.46)

The corresponding equations of motion are given by dx ∂H(x, y) = , dt ∂y or

dx = sgn(y), dt with the initial conditions x(0) = 1,

dx ∂H(x, y) =− , dt ∂x

(3.47)

dy = −sgn(x), dt

(3.48)

y(0) = 0.

(3.49)

Note that in the x-y phase-space the “motion” takes the path (1, 0) → (0, −1) → (−1, 0) → (0, 1) → (1, 0).

(3.50)

The closed path indicates that this is a periodic solution. To determine the explicit solutions, as a function of t, we calculate the solution from (1, 0) to (0, −1), i.e., x(t) goes from x = 1, when t = 0, to x = 0, at time t = t1 ; and y(t) goes from y = 0, at t = 0, to y = −1, when t = t1 . Since in the x-y phase-plane, this motion is in the fourth quadrant, it follows, see Equation (3.48), that the equations of motion are dx = −1, dt

dy = −1. dt

(3.51)

y(t) = C2 − t,

(3.52)

Integrating these equations gives x(t) = C1 − t,

where C1 and C2 are integration constants. Imposing the conditions x(0) = 1,

y(0) = 0,

(3.53)

C2 = 0,

(3.54)

gives C1 = 1, and x(t) = 1 − t,

y(t) = −t.

(3.55)

y(t1 ) = −1,

(3.56)

Now, if we required x(t1 ) = 0,

SQUARE FUNCTIONS

57

then it follows that t1 = 1.

(3.57)

Therefore, we have (

x(t) = 1 − t, y(t) = −t,

for 0 < t < 1.

(3.58)

Continuing this procedure, the full solution can be determined and it is given by the following formulas: ( x(t) = 1 − t, (1, 0) → (0, −1) : for 0 < t < 1. (3.59a) y(t) = −t, ( x(t) = 1 − t, (0, −1) → (−1, 0) : for 1 < t < 2. (3.59b) y(t) = −2 + t, ( x(t) = −3 + t, (−1, 0) → (0, 1) : for 2 < t < 3. (3.59c) y(t) = −2 + t, ( x(t) = −3 + t, (0, 1) → (1, 0) : for 3 < t < 4. (3.59d) y(t) = 4 − t, See Figure 3.5.1 for plots of x(t) and y(t) vs t. From Equations (3.46) and (3.59), we find that x(t) and y(t) have the following properties: (i) They are bounded, i.e., −1 ≤ x(t) ≤ +1,

−1 ≤ y(t) ≤ +1.

(3.60)

(ii) x(t) and y(t) are, respectively, even and odd, i.e., x(−t) = x(t),

y(−t) = −y(t).

(3.61)

(iii) x(t) and y(t) are periodic, with period T = 4. (iv) x(t) and y(t) satisfy the following basic integral relations Z

4

Z

x(t)dt = 0,

0

4

y(t)dt = 0,

(3.62a)

0

Z

4

x(t)y(t)dt = 0,

(3.62b)

0

Z

0

4

x(t)2 dt =

Z

0

4

y(t)2 dt =

4 . 3

(3.62c)

58 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS

x(t) 1.0

0.5

1

2

3

t

4

- 0.5

- 1.0

y(t) 1.0

0.5

0.0

1

2

3

- 0.5

- 1.0

FIGURE 3.5.1: Plots of x(t) and y(t) vs t. See Equations (3.59).

4

t

SQUARE FUNCTIONS

3.6

59

HYPERBOLIC SQUARE FUNCTIONS: DYNAMICS SYSTEM APPROACH

The hyperbolic square functions are solutions to a dynamic system whose Hamiltonian is H(x, y) = |y| − |x| = 1. (3.63) A plot of this function appears in Figure 3.6.1. From dx ∂H dy ∂H = , =− , dt ∂y dt ∂x

(3.64)

we obtain the following equations of motion dx = sgn(y), dt

dy = sgn(x). dt

(3.65)

Close inspection of the results given in Equations (3.65) shows that the flow (in terms of the evolution in time) takes place as in the directions indicated by the arrows in Figure 3.6.2. To summarize this situation, there are two classes of motions for the Hamiltonian given by Equation (3.63): (a) For x(0) = 0 and y(0) = −1, the dynamics evolves on the lower trajectory, with the general motion from right to left. (b) For x(0) = 0 and y(0) = +1, the dynamics takes place on the upper trajectory, with the general motion from left to right. The general, explicit solutions, in terms of t, for these two behaviors are, for −∞ < t < ∞, ( x(t) = t, upper trajectory : (3.66a) y(t) = 1 + |t|, ( x(t) = −t, lower trajectory : (3.66b) y(t) = −(1 + |t|). Figure 3.6.3 plots these functions with time. Finally, observe that x(t) is an odd function, while y(t) is an even function of t.

60 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS

4

2

-4

-2

2 -2

-4

FIGURE 3.6.1: Plot of H(x, y) = |y| − |x| = 1.

4

SQUARE FUNCTIONS

61

4

2

-4

-2

2

4

-2

-4

FIGURE 3.6.2: The flow along the trajectories in the x-y phase-space for H(x, y) = |y| − |x| = 1. See Equations (3.65).

62 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS 1.0

0.5

- 1.0

- 0.5

0.5

1.0

- 0.5

- 1.0

5

1

-

-

0

FIGURE 3.6.3a: Plot of x(t) and y(t) for upper trajectory where x(0) = 0 and y(0) = 1.

SQUARE FUNCTIONS

63

x(t) 1.0

0.5

t - 1.0

- 0.5

0.5

1.0

2

4

- 0.5

- 1.0

y(t) -4

-2

t

-1

-2

-3

-4

-5

FIGURE 3.6.3b: Plot of x(t) and y(t) for lower trajectory where x(0) = 0 and y(0) = −1.

64 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS

2

1

- 1.0

- 0.5

0.5

1.0

-1

-2

FIGURE 3.7.1: Definition of r and θ for the periodic hyperbolic square functions.

3.7

PERIODIC HYPERBOLIC SQUARE FUNCTIONS

For this situation, we have |y| − |x| = 1 2

2

2

y +x =r ,

(3.67a) (3.67b)

with x(θ) = r(θ) cos θ,

y(θ) = r(θ) sin θ.

(3.68)

See Figure 3.7.1 for the definitions of r and θ. If Equations (3.68) are substituted into Equation (3.67a), then the following result is obtained |r sin θ| − |r cos θ| = r[|sin θ| − |cos|] = 1, and solving for r gives r(θ) =

(3.69)

1 . |sin θ| − |cos θ|

(3.70)

r(θ) > 0,

(3.71)

Inspection of the expression for r(θ) shows that the requirement

SQUARE FUNCTIONS

65

permits only the following allowed angles π 3π 5π 7π 0, 0 ≤ θ ≤ 2π, r(−θ) = r(θ),   r(θ + π) = r(θ),

(4.24)

(4.25)

and it follows that x(θ) and y(θ) are, respectively, even and odd, i.e., x(−θ) = x(θ),

y(−θ) = −y(θ),

(4.26)

PARABOLIC TRIGONOMETRIC FUNCTIONS

73

and are periodic, with period T = 2π, i.e., x(θ + 2π) = x(θ),

y(θ + 2π) = y(θ).

(4.27)

The function r(θ) can be determined by substituting the expressions in Equation (4.24) into Equation (4.22); doing this gives (cos θ)2 r2 + (2|sin θ|)r − 1 = 0.

(4.28)

This latter equation has, for a given θ, two solutions. Since r(θ) > 0, we select the solution satisfying this condition and it is  h i p 1 2 + 4(cos θ)2 r(θ) = −2|sin θ| + 4|sin θ| 2(cos θ)2 1 − |sin θ| = , (4.29) (cos θ)2 where the latter equation can also be written as ( 1−sin θ θ)2 , 0 ≤ θ ≤ π, r(θ) = (cos 1+sin θ (cos θ)2 , π ≤ θ ≤ 2π. Using (cos θ)2 = 1 − (sin θ)2 , Equation (4.30) can be rewritten as ( 1 θ , 0 ≤ θ ≤ π, r(θ) = 1+sin 1 1−sin θ , π ≤ θ ≤ 2π. Note that

(


r(0) = 1, r π2 = 12 , r(π) = 1
1 r 3π = 2 , r(2π) = 1. 2

The parabolic trigonometric functions are defined as follows:   pbd(θ) ≡ r(θ), pbc(θ) ≡ r(θ) cos θ,   pbs(θ) ≡ r(θ) sin θ,

(4.30)

(4.31)

(4.32)

(4.33)

where they are “called,” respectively, the parabolic dine, cosine, and sine functions. Graphs of these three functions are given in Figure 4.3.2.

4.4

|y| −

  1 2

x2 =

1 2

AS A DYNAMIC SYSTEM

Consider the Hamiltonian H(x, y) = |y| −

  1 1 x2 = , 2 2

(4.34)

74 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS

pbd(

)

1.0

0.9

0.8

0.7

0.6

2

1

1.0

pbc(

4

3

5

6

)

0.5

1

2

3

4

5

6

2

3

4

5

6

- 0.5

- 1.0

pbs(

)

0.4

0.2

1 - 0.2

- 0.4

FIGURE 4.3.2: Graphs of the parabolic trigonometric functions.

PARABOLIC TRIGONOMETRIC FUNCTIONS which gives rise to following equations of motion   dx = ∂H(x,y) = sgn(y), dt

 dy dt

∂y

=

− ∂H(x,y) ∂x

75

(4.35)

= x.

Two possible sets of initial conditions are A : x(0) = 0, B : x(0) = 0,

1 2  1 y(0) = − . 2 y(0) =

(4.36) (4.37)

As Figure 4.4.1 indicates, initial conditions A and B, respectively, correspond to motions along the upper and lower phase-space trajectories, i.e.,   1 1 y>0:y= + x2 , (4.38) 2 2     1 1 y 0, will now be calculated. For this case ( dx dt = +1, x(0) = 0, (4.40) dy y(0) = 12 , dt = x, and the solutions are given by the expressions   1 1 2 x(t) = t, y(t) = + t . 2 2 Likewise, for initial conditions B, where y < 0, we have ( dx dt = −1, x(0) = 0,
dy y(0) = − 12 , dt = x, with solutions

x(t) = −t,

    1 1 2 y(t) = − − t . 2 2

(4.41)

(4.42)

(4.43)

If we extend t to the interval, −∞ < t < ∞, then x(t) and y(t) are, respectively, odd and even functions of t. These functions are plotted in Figure 4.4.2. Observe that both x(t) and y(t) are unbounded.

76 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS

2

1

-2

-1

1

2

-1

-2

FIGURE 4.4.1: Graphs of H(x, y) = |y| − tions of the phase-space flow.

1 2




x2 = 12 , along with the direc-

PARABOLIC TRIGONOMETRIC FUNCTIONS

77

1.0

0.5

- 1.0

- 0.5

0.5

1.0

0.5

1.0

- 0.5

- 1.0

1.0

0.8 0.6 0. 0.

- 1.0

- 0.5

0.0

FIGURE 4.4.2a: x(t) and y(t) vs t for x(0) = 0 and y(0) = − 21 , and −∞ < t < ∞.

78 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS 1.0

0.5

- 1.0

- 0.5

0.5

1.0

- 0.5

- 1.0

y(t) -4

x

-2

2 -

4

1 2

-1

-

3 2

FIGURE 4.4.2b: x(t) and y(t) vs t for x(0) = 0 and y(0) = 12 , and −∞ < t < ∞.

4.5

GEOMETRIC ANALYSIS OF |y| −

  1 2

The parabolic hyperbolic-type functions associated with   1 1 x2 = , |y| − 2 2

x2 =

1 2

(4.44)

PARABOLIC TRIGONOMETRIC FUNCTIONS

79

1.0

0.5

- 1.0

- 0.5

0.5

1.0

- 0.5

- 1.0

FIGURE 4.5.1: Definition of r and θ for the curve |y| − will now be examined; see Figure 4.5.1. We have  2 2 2  r (θ) = x(θ) + y(θ) , x(θ) = r(θ) cos θ,   y(θ) = r(θ) sin θ.

1 2




x2 = 12 .

(4.45)

Substituting x(θ) and y(θ) into Equation (4.44) gives the following expression for r(θ) (cos θ)2 r2 − (2|sin θ|)r + 1 = 0. (4.46) Inspection of Equation (4.46) indicates that it has either two positive roots or a pair of complex conjugate roots. Solving this quadratic equation for r(θ) gives  h i p 1 2 − (cos θ)2 r± (θ) = |sin θ| ± |sin θ| (cos θ)2 p |sin θ| ± (− cos 2θ) = , (4.47) (cos θ)2

80 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS where |sin θ|2 − (cos θ)2 = (sin θ)2 − (cos θ)2     1 1 = (1 − cos 2θ) − (1 + cos 2θ) 2 2 = − cos 2θ,

(4.48)

was used to simplify the item within the square root. Observe that the results in Equation (4.47) allow the following conclusions to be reached:
(i) The ray from the origin can intersect the curve |y| − 21 x2 = 12 , no where, once, or twice. (ii) In the following angular intervals, rays from the origin do not intersect the curve 0≤θ
0,

(5.19)

86 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS see Equation (5.4). This implies that d(t) should be selected such that d(t) > 0,

d(−t) = d(t).

(5.20)

In other words, d(t) is a non-negative, even function. We name d(t) the “dine” function associated with c(t) and s(t). Squaring the two expressions appearing in Equations (5.18) and adding gives  2  2 dc ds + = d2 (c2 + s2 ) = d2 , (5.21) dt dt and this allows the determination of d(t) in terms of the derivative of c(t) and s(t); carrying out this calculation provides the expression s  2  2 ds dc + . (5.22) d(t) = dt dt Note that for the case where c(t) and s(t) are periodic, d(t) is also periodic. A nontrivial illustration of this is the Jacobi elliptic functions: cn(t), sn(t), and dn(t).

5.3

MATHEMATICAL STRUCTURE OF θ(t)

In addition to being odd, what other restrictions must be placed on θ(t) to ensure that c(t) and s(t) are periodic? Now, the periodicity conditions for c(t) and s(t) are the requirements c(t + T ) = c(t),

s(t + T ) = s(t),

(5.23)

where T > 0 is the period. This means that eiθ(t+T ) = c(t + T ) + is(t + T ) = c(t) + is(t) = eiθt .

(5.24)

Using 1 = e2πki ,

k = integer,

(5.25)

it follows that eiθ(t+T ) = 1 · eiθ(t) = e2πki · eiθ(t) = ei[θ(t)+2πk] .

(5.26)

Taking the “minimum period” corresponds to selecting k = 1; doing this gives θ(t + T ) = θ(t) + 2π,

(5.27)

GENERALIZED PERIODIC SOLUTIONS OF f (t)2 + g(t)2 = 1

87

which is a first-order, linear inhomogeneous difference equation. The general solution to this equation is   2π θ(t) = A(t) + t, (5.28) T where A(t) has the properties A(−t) = A(t),

A(t + T ) = A(t).

(5.29)

Since θ(t) is assumed to have a continuous, second derivative, this condition applies also to A(t). One consequence of this analysis is that A(t) has a sinetype Fourier representation, i.e., A(t) =

∞ X

ak sin

k=1



2πkt T



,

(5.30)

where the coefficients satisfy the upper bound M , k3

|ak | ≤

(5.31)

where the positive constant, M , depends on A(t). Note that θ(t) is not periodic. In fact, we have   2π θ(t + T ) = A(t + T ) + (t + T ) T   2π t + 2π = A(t) + T = θ(t) + 2π. (5.32) From Equations (5.11), we have   dc(t) = − dt

 ds(t) = dt

dθ dt


dθ dt




sin θ(t),

(5.33)

cos θ(t),

and using these expressions in Equation (5.22) gives the result s s  2  2  2 dθ ds dc dθ + = [(cos θ)2 + (sin θ)2 ] = . d(t) = dt dt dt dt

(5.34)

However, without loss of (too much) generality, we will restrict the function, θ(t), such that dθ(t) > 0. (5.35) d(t) = dt

88 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS Thus, it follows from Equation (5.28) that d(t) =

dA(t) 2π + . dt T

(5.36)

Since d(t) > 0, the derivative of A(t) satisfies the condition   dA(t) 2π . >− dt T

(5.37)

If we had chosen to use the absolute value in the determination of d(t), i.e., dθ(t) , d(t) = (5.38) dt

then no restriction would be placed on dA(t)/dt. In the following, we will assume that the conditions in Equations (5.36) and (5.37) hold.

5.4

AN EXAMPLE: A(t) = a1 sin



2π T



t

The simplest mathematical structure for A(t) is one for which only a single harmonic is included, i.e., A(t) has the form   2π t, (5.39) A(t) = a1 sin T and θ(t) is given by the expression θ(t) + a1 sin



2π T



t+



2π T



t,

and the corresponding c(t), s(t), and d(t) are       2π 2π c(t) = cos a1 sin t+ t , T T       2π 2π s(t) = sin a1 sin t+ t , T T      2π 2π 1 + a1 cos t . d(t) = T T

(5.40)

(5.41) (5.42) (5.43)

The restriction d(t) > 0, requires that |a1 | < 1.

(5.44)

GENERALIZED PERIODIC SOLUTIONS OF f (t)2 + g(t)2 = 1

89

It turns out that explicit expressions can be found for the Fourier series of c(t) and s(t). These results are based on the following two formulas cos(z sin t) = J0 (z) + 2

∞ X

J2k (z) cos(2kt),

(5.45a)

k=1

sin(z cos t) = 2

∞ X

J2k+1 (z) sin(2k + 1)t,

(5.45b)

k=0

and the trigonometric relations cos[φ1 (t) + φ2 (t)] = cos φ1 (t) cos φ2 (t) − sin φ1 (t) sin φ2 (t), sin[φ1 (t) + φ2 (t)] = sin φ1 (t) cos φ2 (t) + cos φ1 (t) sin φ2 (t).

(5.46a) (5.46b)

These Fourier series are (for T = 2π) c(t) = cos(t + a1 sin t) ∞ X = −J1 (a1 ) + [J2k−1 (a1 ) − J2k+1 (a1 )] cos(2kt) k=1

+

∞ X

[J2k (a1 ) + J2k+2 (a1 )] cos(2k + 1)t,

(5.47)

k=0

s(t) = sin(t + a1 sin t) ∞ X = [J2k−1 (a1 ) + J2k+1 (a1 )] sin(2kt) k=1 ∞ X

+

[J2k (a1 ) + J2k+2 (a1 )] sin(2k + 1)t.

(5.48)

k=0

In the above relations, Jk (z) is the k-th order Bessel function. Figures 5.4.1 to 5.4.2 are illustrative plots of θ(t), c(t), s(t), and d(t).

5.5

DIFFERENTIAL EQUATION FOR f (t) AND g(t)

The functions f (t) and g(t) are solutions to the same linear, second-order, non-autonomous differential equation. To demonstrate this, we start with df = −dg, dt

dg = df. dt

(5.49)

90 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS

Θ (t) = t + [0.2] sin (t) 25 20 15 10 5 t 5

10

15

20

25

10

15

20

25

d(t) = 1 + 0.2 cos(t) 1.2 1.1 1.0 0.9 t 5

FIGURE 5.4.1: Plots of θ(t) vs t and d(t) vs t, for a1 = 0.2.

GENERALIZED PERIODIC SOLUTIONS OF f (t)2 + g(t)2 = 1

c(t) = cos(0.2 sin(t) + t) 1.0 0.5 t 5

10

15

20

25

10

15

20

25

-0.5 -1.0

s(t) = sin(0.2 sin(t) + t) 1.0 0.5 t 5 -0.5 -1.0 FIGURE 5.4.2: Plots of c(t) and s(t).

91

92 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS This is just a relabeling of Equations (5.18). Taking the derivative of the first expression gives  ′  ′ f d ′′ ′ ′ ′ f = −d g − dg = (−d ) − d(df ) = f ′ − d2 f. (5.50) −d d Since d(t) = θ′ (t), we finally obtain  ′ d f ′′ − f ′ + d2 f = 0. d

(5.51)

Starting with the second expression in Equation (5.49), we, likewise, obtain  ′ d g ′′ − g ′ + d2 g = 0. (5.52) d Note that Equations (5.51) and (5.52) are the same differential equation, except for the label placed on the function. Therefore, this situation can be summarized as follows:  ′ ( f (t) : w′′ − dd w′ + d2 w = 0, (5.53) w(0) = 1, w′ (0) = 0; (

g(t) : w′′ −

 ′ d d

w(0) = 0,

w′ + d2 w = 0, w′ (0) = 1.

Making use of the fact that d(t) = θ′ (t), we have  ′′  θ ′′ w′ + (θ′ )2 w = 0. w − θ′

5.6

(5.54)

(5.55)

DISCUSSION

We now summarize and discuss what has been obtained in this chapter: i) The functional equation, f 2 (t) + g 2 (t) = 1, has a class of T -periodic solutions that can be expressed as θ(t) = cos θ(t),

s(t) = sin θ(t),

(5.56)



(5.57)

where θ(t) has the form θ(t) = A(t) +



2π T

t,

GENERALIZED PERIODIC SOLUTIONS OF f (t)2 + g(t)2 = 1 with A(t) =

∞ X

ak sin

k=1



2πk T



t.

93

(5.58)

ii) A knowledge of θ(t) allows for the complete determination of the three periodic functions: c(t), s(t), and d(t) = θ′ (t). iii) c(t) and d(t) are even functions of t, while s(t) is an odd function. These results are a consequence of θ(t) being an odd function of t. iv) Given θ(t), then f (t) and g(t) satisfy the following coupled, first-order differential equations df (t) = −θ′ (t)g(t), dt

dg(t) = θ′ (t)f (t), dt

and the following second-order equation  2    ′′  d f (t) θ (t) d ′ 2 = 0. − + [θ (t)] dt2 θ′ (t) dt g(t)

(5.59)

(5.60)

Examination of Equation (5.1) shows that it is invariant under t → −t, and such a condition also holds for the differential Equation (5.60). Since this equation is linear, the previous result implies that its solutions can be constructed to be either even or odd functions of t. Observe that if θ(t) = t, then Equation (5.60) becomes  2   d f (t) +1 = 0, (5.61) dt2 g(t) with solutions f (t) = cos(t),

g(t) = sin(t).

(5.62)

v) The Jacobi elliptic functions, (cn, sn, dn), depend not only on t, but also a parameter k, where 0 ≤ k ≤ 1. However, in the discussion to follow, its functional dependencies, with regard to the Jacobi elliptic functions, are not important. Assume that θ′ (t) = dn(t), (5.63) then it follows that θ(t) =

Z

t

dn(z)dz,

(5.64)

0

and

eiθ(t) = C(t) + iS(t) = cos θ(t) + i sin θ(t).

(5.65)

Therefore, C(t) = cos S(t) = sin

Z

t

0 Z t 0



dn(z)dz ,  dn(z)dz ,

(5.66a) (5.66b)

94 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS and taking the respective derivatives of C(t) and S(t) gives dC(t) = −dn(t)S(t), dt dS(t) = dn(t)C(t), dt

(5.67a) (5.67b)

and these are the coupled, first-order differential equations for the Jacobi elliptic functions, i.e., C(t) = cn(t),

S(t) = sn(t).

(5.68)

Thus, we conclude that the Jacobi cosine and sine functions are the real and imaginary parts of the complex function, exp[iθ(t)], where θ(t) is related to the Jacobi dine function through the relationship expressed in Equation (5.64). This means that cn(t) and sn(t) are solutions to both a nonlinear, autonomous, second-order differential equation having a cubic nonlinearity and also solutions to a linear, non-autonomous, second-order differential equation.

5.7 is

NON-PERIODIC SOLUTIONS OF f 2 (t) + g 2 (t) = 1

Equation (5.1) has non-periodic solutions. An example for −∞ < t < +∞, f (t) =

(

1, t = rational number, 0, t = irrational number;

(5.69a)

g(t) =

(

0, t = rational number, 1, t = irrational number.

(5.69b)

Other elementary examples are (−∞ < t < +∞) f (t) = 1, and

(

f (t) =

√1 , 2

g(t) = 0,

g(t) =

√1 , 2

f (t) = −1, g(t) = 0,

for t < 0; for t > 0.

(5.70)

(5.71)

Note that the functions, f (t) and g(t), are not continuous at t = 0. A more interesting set of functions are those that are oscillatory, but not periodic. So, taking θ(t) = t2 , (5.72) we have eiθ(t) = cos(t2 ) + i sin(t2 ),

(5.73)

GENERALIZED PERIODIC SOLUTIONS OF f (t)2 + g(t)2 = 1

95

cost2  1.0

t 2

10

2

10

-1.0

sint + t2  1.0

t -1.0 FIGURE 5.7.1: Two oscillatory, non-periodic functions, corresponding to θ(t) = t2 and θ(t) = t + t2 .

96 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS

t2 cos 1 + t2 1.0 0.9

t 2

10

12

2

10

12

t2 sin 1 + t2

t FIGURE 5.7.2: Two oscillatory, non-periodic functions, corresponding to θ(t) = t2 /(1 + t2 ).

GENERALIZED PERIODIC SOLUTIONS OF f (t)2 + g(t)2 = 1

97

and f (t) = cos(t2 ),

g(t) = sin(t2 ).

(5.74) 2

The effective period at time t can be calculated. Start with t and write it as follows 2π t2 = · t, (5.75) T (t) where T (t) is the effective period. Therefore, T (t) is T (t) =

2π , t

(5.76)

and we find that T (t) decreases with T . The results shown in Figures 5.7.1 and 5.7.2 illustrate this phenomena, for θ(t) = t2 and θ(t) = t + t2 .

NOTES AND REFERENCES Section 5.1: This chapter is based on work presented in a previous article: R. E. Mickens, Periodic solutions of the functional equation f (t)2 + g(t)2 = 1, Journal of Difference Equations and Applications 22 (2016), 67–74. Section 5.2: Starting from c2 + s2 = 1, we have the result c

dc ds +s =0 dt dt

on taking the derivative. If we wish to split this into two expressions, one each for dc/dt and ds/dt, then a possibility is dc = −d(t)s, dt

ds = d(t)s. dt

The simplest choice is that d(t) is a constant. But, as we show in this section, the more interesting and more productive selection is having d be a function of t. It should be noted that in Hamiltonian dynamics d(t) is taken to be one. Section 5.3: Methods to solve the first-order difference Equation (5.27) are given in R. E. Mickens, Difference Equations: Theory, Applications and Advanced Topics, 3rd edition (CRC Press; Boca Raton, FL; 2015), Section 2.2. The results expressed in Equations (5.30) and (5.31) may be justified using techniques from the theory of Fourier analysis. See, for example: R. E. Mickens, Mathematical Methods for the Natural and Engineering Sciences (World Scientific, London, 2004), Section 2.5. Section 5.4: The mathematical relations, given in Equations (5.45) are from I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, New York, 1965). See formulas 8.511.3 and 8.511.4.

Chapter 6 RESUMÉ OF (SOME) PREVIOUS RESULTS ON GENERALIZED TRIGONOMETRIC FUNCTIONS

6.1

INTRODUCTION

This chapter provides a very brief summary of work on generalized trigonometric functions and related topics. We do not seek completeness in terms of all the various formulations and results which have appeared in the scientific and mathematical research literature. Upon examining these works, it becomes clear that there is not full unity with regard to the definition of the generalized sine and cosine functions. This is also demonstrated in the present work of the author within this book. Consequently, the main task of the chapter is to introduce and summarize some of the publications that have influenced my views of generalized trigonometric functions. Section 6.2 approaches the generalized sine and cosine functions by defining them as solutions to two coupled, nonlinear, differential equations, with specified initial conditions. Many physical systems are formulated this way, in particular, systems which can be modeled by Hamiltonians. Section 6.3 introduces the generalized sine and cosine by way of an integral representation. This way of proceeding is similar to how the standard sine and cosine functions are “constructed” in a calculus course. Section 6.4 is devoted to defining generalized sine and cosine functions in terms of the geometric properties of curves in a two-dimensional plane. We show that in addition to the usually expected sine- and cosine-type functions, there is an associated third periodic function, the “dine.” This origin of the dine function was explored in all of the previous five chapters. Section 6.5 is not directly connected to most of the published articles on the generalized sine and cosine functions, discussed in Sections 6.2, 6.3, and 6.4. However, the use of symmetry considerations has played an important role in the work of the author, and in this section, we show how the invariances of the phase-space curves can be used to deduce very strict constraints on the properties of the generalized sine and cosine functions. Finally, Section 6.6 provides a general, but brief discussion of the sinpq and

99

100 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS cospq functions, and their relationship to the standard trigonometric sine and cosine functions.

6.2

DIFFERENTIAL EQUATION FORMULATION

The standard trigonometric sine and cosine functions may be defined by means of the following system of coupled, first-order, linear differential equations  dx(t)   dt = −y(t), dy(t) (6.1) = x(t),   dt x(0) = 1, y(0) = 0, where

x(t) = cos t,

y(t) = sin t.

(6.2)

Note also that both x(t) and y(t) satisfy the following set of second-order, linear differential equations d2 x(t) + x(t) = 0, dt2 2 d y(t) + y(t) = 0, dt2

x(0) = 1, y(0) = 0,

dx(0) = 0, dt dy(0) = 1. dt

(6.3a) (6.3b)

Also, if the first and second expressions in Equation (6.1) are multiplied, respectively, by x(t) and y(t), and integrated, then we obtain x(t)2 + y(t)2 = 1,

(6.4)

which corresponds to the Pythagorean identity. These results may be generalized (Shelupsky, 1959) by considering the following system of coupled, nonlinear differential equations  dx(t)   dt = −f (y), dy(t) (6.5) = f (x),   dt x(0) = 1, y(0) = 0, where

f (z) = |z|p−2 z, Since f (x)

dx = −f (x)f (y), dt

p > 1.

f (y)

dy = f (y)f (x), dt

(6.6) (6.7)

it follows that |x(t)|p + |y(t)|p = 1.

(6.8)

PREVIOUS RESULTS

101

If we now make the definitions x(t) ≡ sinp (t),

y(t) = cosp (t),

then the following generalized Pythagorean identity is obtained |sinp (t)|p + |cosp (t)|p = 1.

(6.9)

Thus, this definition of the generalized sine and cosine function is based on the differential Equations (6.5), which in turn gives rise to the first-integral presented in Equation (6.9). The above arguments can be extended to cover the situation for which the corresponding sine function depends on two parameters, p and q. This gives rise to the identity |sinpq (t)|p + |cospq (t)|q = 1, (6.10) which can be used to define cospq (t), i.e., 1/q

cospq (t) = (±) [1 − |sinpq (t)|p ]

6.3

.

(6.11)

DEFINITION AS INTEGRAL FORMS

Generalized trigonometric functions may also be defined in terms grals. This method is based on an extension of the definition of the sine function, i.e., Z x dz √ , 0 ≤ |x| ≤ 1, t= 1 − z2 0 and x(t) = sin t,

of interegular (6.12) (6.13)

to the expression sin−1 p ≡

(R t

dz , 0 (1−z p )1/p R −t (−) 0 (1−zdzp )1/p ,

0 ≤ t ≤ 1,

−1 ≤ t ≤ 0,

(6.14)

where cosp (t) is related to sinp (t) by the relationship |sinp (t)|p + |cosp (t)|p = 1.

(6.15)

Two parameter generalizations have been constructed for the sine function. An example of such an extension is (R t dz 0 (1−z q )1/p , 0 ≤ t ≤ 1, −1 R −t sinpq (t) ≡ (6.16) (−) 0 (1−zdzq )1/p , −1 ≤ t ≤ 0,

102 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS with |sinpq (t)|p + |cospq (t)|q = 1.

(6.17)

Generally, the (p, q) are taken as positive. However, particular requirements may force p > 1 and q > 1.

6.4

GEOMETRIC APPROACH

In Chapters 1 to 4 of this book, we have investigated the construction of periodic solutions to the functional equation |y|p ± |x|q = 1,

(6.18)

where x and y are functions of an angle θ, i.e., x = x(θ),

y = y(θ),

0 ≤ θ ≤ 2π.

(6.19)

This general technique was also used by Mahdi et al. (2014) to study the properties of functions x(θ) and y(θ) satisfying the constraint |x|m + |y|m = 1,

m > 0.

(6.20)

As we have shown, the derived functions are always periodic with period 2π, but some of the periodic solutions may be unbounded. Considering the relation in Equation (6.20), x and y are replaced by x(θ) = r(θ) cos θ,

y(θ) = r(θ) sin θ.

(6.21)

Therefore, substitution of these representations into Equation (6.20) and solving for r(θ) gives 1 r(θ) = , (6.22) m [|cos θ| + |sin θ|m ]1/m

and we have

x(θ) ≡ cosm (θ) =

cos θ , [|cos θ|m + |sin θ|m ]1/m

y(θ) ≡ sinm (θ) =

[|cos θ|m

r(θ) ≡ dinm (θ) =

1 . [|cos θ|m + |sin θ|m ]1/m

sin θ , + |sin θ|m ]1/m

(6.23) (6.24) (6.25)

Note first that there are three basic periodic functions. Second, cosm (θ) and sinm (θ) have a period 2π, the related dine function, dinm (θ), has a period of π.

PREVIOUS RESULTS

6.5

103

SYMMETRY CONSIDERATIONS AND CONSEQUENCES

This section gives a brief summary of results which can be derived from the application of symmetry transformations to the formula ( |y|p + |x|q = 1, (p ≥ 1, q ≥ 1), (6.26) x(0) = 1, y(0) = 0, where x = x(t) and y = y(t). This means that Equation (6.26) is taken to represent a dynamic system in the x-y phase-plane, and t represents the time.

6.5.1

Symmetry Transformation and Consequences

Consider the following transformation of the x and y variables: T1 : x → −x, y → y, T2 : x → x, y → −y,

(6.27a) (6.27b)

T3 = T1 T2 = T2 T1 : x → −x,

y → −y.

(6.27c)

T1 corresponds to reflection in the y-axis; T2 to reflection in the x-axis; and T3 to inversion through the origin. For H(x, y) ≡ |y|p + |x|q , we have

  T1 H(x, y) = H(−x, y) = H(x, y), T2 H(x, y) = H(x, −y) = H(x, y),   T3 H(x, y) = H(−x, −y) = H(x, y).

(6.28)

(6.29)

Thus, it can be concluded that H(x, y) is invariant under the three transformations (T1 , T2 , T3 ). This fact allows the following conclusions to be made: (a) Equation (6.26) is a simple, closed, convex curve in the x-y phase-plane. (b) Considered as a dynamic system, x(t) and y(t) are periodic, i.e., there exists a Tpq such that x(t + Tpq ) = x(t),

y(t + Tpq ) = y(t).

(6.30)

(Tpq will be calculated in Section 6.5.3.) (c) x(t) and y(t) satisfy the bounds −1 ≤ x(t) ≤ 1,

−1 ≤ y(t) ≤ 1.

(6.31)

(d) x(t) and y(t) are, respectively, even and odd functions of t, i.e., x(−t) = x(t),

y(−t) = −y(t).

(6.32)

104 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS

6.5.2

Hamiltonian Formulation

Let H(x, y) be considered the Hamiltonian of a dynamic system; see Equation (6.28). The equations of motion are dx ∂H(x, y) = , dt ∂y

dy ∂H(x, y) =− , dt ∂x

(6.33)

and from these relations the following expressions result dx = p|y|p−1 sgn(y), dt dy = −q|x|q−1 sgn(x), dt where sgn(z) is the “sign” function, i.e.,   +1 z sgn(z) = 0 z   −1 z

x(0) = 1,

(6.34)

y(0) = 0,

(6.35)

> 0, = 0, < 0.

(6.36)

The first-order differential equation for the trajectories in the x-y phasespace, y = y(x), is given by the equation    dy q|x|q−1 sgn(x) = (−) . (6.37) dx p|y|p−1 sgn(y) y

(1,0) x

FIGURE 6.5.1: Plot of the phase-space trajectory in the fourth-quadrant beginning at (1, 0) and ending at (0, −1). The time to do this is Tpq /4.

PREVIOUS RESULTS

6.5.3

105

Area of Enclosed Curve

Let us now calculate the area interior to the closed curve represented in Equation (6.26). Note that because of the invariance under the symmetry transformations, (T1 , T2 , T3 ), this area is four times the area included in the first quadrant of the x-y plane, i.e., area = A = 4

Z

1

y(x)dx,

(6.38)

1/p

(6.39)

0

where

y(x) = + [1 − |x|q ] Therefore, A=4

Z

0

Now, let

1

.

[1 − |x|q ]1/p dx.

u = xq ⇒ x = u1/q ⇒ dx =

  1−q 1 u q du. q

(6.40)

(6.41)

With this change of variable, Equation (6.40) becomes A=

 Z 1 1−q 4 u q (1 − u)1/p du. q 0

Using the definition of the beta function Z 1 Γ(a)Γ(b) , B(a, b) = ua−1 (1 − u)b−1 = Γ(a + b) 0 where Γ(z) is the gamma function, we obtain for the area the result      Γ 1 Γ 1+ 1 q p 4  .  A= q Γ 1 + q1 + p1

(6.42)

(6.43)

(6.44)

Using the relationship

Γ(z + 1) = zΓ(z),

(6.45)

we finally have A=



4 pq



     Γ 1p Γ q1    , Γ 1 + 1p + q1 

(6.46)

and the area enclosed by the curve |y|p + |x|q = 1. Observe that the area, A = A(p, q), is symmetric with respect to p and q, i.e., A(p, q) = A(q, p). (6.47)

106 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS

6.5.4

Period

The time taken to travel from (1, 0) to (0, −1) in phase-space, is one-fourth of the period Tpq . This result is a consequence of the symmetry properties of the phase-space trajectories; see the discussion presented in Section 6.5.1. Along this portion of the curve 1/p

y(x) = (−1) [1 − |x|q ]

,

x ≥ 0.

(6.48)

Therefore,

p−1 dx = p|y|p−1 sgn(y) = (−)p [1 − |x|q ] p , dt and it follows that   1 dx −dt = , p [1 − xq ] p−1 p

and

−

Z

Tpq /4

0

Therefore Tpq =

(6.49) (6.50)

 Z 0 1 dx dt = p−1 . p 1 [1 − xq ] p

(6.51)

 Z 1 1−p 4 [1 − xq ] p dx, p 0

(6.52)

which with the transformation of variable u = xq

(6.53)

becomes Tpq =



4 pq

Z

1 0

(1 − u)

1−p p

u

1−q q

du =



      Γ 1 Γ 1 p q 4    . 1 1 pq Γ p+q

(6.54)

Note that, just as for the area formula, Tpq = T (p, q) is symmetric in p and q, i.e., T (p, q) = T (q, p). (6.55) From the above analysis, it follows that x(t) and y(t), which we can denote as x(t) = cospq (t), have the following special values:  x(0)    =1   T  x pq = 0     4  T x 2pq = −1      3Tpq  x =0  4    x(Tpq ) = 1

y(t) = sinpq (t),

(6.56)

y(0)=0   T y 4pq = −1   T y 4pq = 0   3Tpq y =1 4

(6.57)

y(Tpq ) = 0

PREVIOUS RESULTS

6.6

107

SUMMARY

The expressions in Equations (6.10), (6.18), and (6.26) may be used to define generalized trigonometric sine and cosine functions. They differ in what is considered the independent variable for the x and y functions. Of importance is the fact that the general relation ( |y|p + |x|q = 1, x(0) = 1, y(0) = 0, (6.58) p ≥ 1, q ≥ 1, is invariant under the transformation stated in Equation (6.29). Since the standard sine and cosine trigonometric functions correspond to the case where p = q = 2, it is not at all surprising that the generalized sine and cosine functions have similar properties, as reflected in the results stated in Sections 6.5.1 and 6.5.4.

NOTES AND REFERENCES Section 6.1: An excellent introduction to “sine” functions is the book listed below. It covers the generalized sine, as taken by the author to be the inverse function defined by an integral of the form Z y dz √ x= . 1 + mz 2 + nz 4 0 The four cases considered are • circular sine: m = −1, n = 0; • hyperbolic sine: m = 1, n = 0; • lemniscate sine: m = 0, n = −1; • sine amplitude of Jacobi: m = −(1 + k 2 ), n = k 2 , for 0 < k < 1. (1) A. I. Markushevich, The Remarkable Sine Function (Elsevier, New York, 1966). Section 6.2: The topics of this section are discussed in the references: (1) D. Shelupsky, A generalization of the trigonometric functions. The American Mathematical Monthly 66 (1959), 879–884.

108 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS (2) D. Wei, Y. Liu, and M. B. Elgindi, Some generalized trigonometric sine functions and their applications. Applied Mathematical Sciences 6 (2012), 6053–6068. (3) R. M. Rosenberg, The Ateb(h)-functions and their properties. Quarterly of Applied Mathematics 21 (1963), 37–47. Section 6.3: In addition to ref. [2], given under the items in Section 6.2 above, see also the following: (1) J. Lang and D. Edmunds, Eigenvalues, Embeddings and Generalised Trigonometric Functions (Springer, Heidelberg, 2011), Section 2.1. Section 6.4: A good summary of the “purely” geometric approach is given in the paper: (1) H. Mahdi, M. Elatrash, and S. Elmadhoun, On generalized trigonometric functions. Journal of Mathematical Sciences and Applications 2 (2014), 33–38. Section 6.5: The geometry of two-dimensional (x − y) phase-space is discussed in the following books: (1) M. Humi and W. Miller, Second Course in Ordinary Differential Equations for Scientists and Engineers (Springer-Verlag, New York, 1988). See Chapter 8. (2) R. E. Mickens, Mathematical Methods for the Natural and Engineering Sciences, 2nd Edition (World Scientific, London, 2017). See Sections 4.4 and 4.5.

Chapter 7 GENERALIZED TRIGONOMETRIC FUNCTIONS: |y|p + |x|q = 1

7.1

INTRODUCTION

This chapter presents and discusses a method for determining periodic solutions to the functional equation |y|p + |x|q = 1,

p > 0,

q > 0.

(7.1)

We show that functions x(t) and y(t) can be selected such that x(t + T ) = x(t),

y(t + T ) = y(t)

(7.2)

for any T > 0. Further, our procedure constructs x(t) and y(t) that mimic the standard trigonometric cosine and sine functions, i.e., x(t) and y(t) are, respectively, even and odd, and bounded. Further, x(t) and y(t) have zeros and maximum/minimum values at the t values (0, T /4, T /2, 3T /4, T ). Section 7.2 outlines the general methodology, while the explicit functions are given in Section 7.3. Finally, in Section 7.4, we present a gallery of plots of x(t) and y(t) for a selected set of (p, q) values.

7.2

METHODOLOGY

Our purpose is to construct explicit solutions to Equation (7.1) such that x(t) and y(t) have exactly the same essential properties as the standard cosine and sine functions. This means that the following hold: (i) x(0) = 1,

y(0) = 0.

(7.3)

(ii) x(−t) = x(t)

y(−t) = −y(t).

(7.4) 109

110 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS (iii) For given T > 0, the periodicity condition of Equation (7.2) holds. (iv) At the indicated specific values of t, we have (

x(0) = 1, x T4 = 0, x T2 = −1,
x 3T = 0, x(T ) = 1, 4 (



T y(0) = 0, y T2 = 0, 4 = 1,
y 3T = −1, y(T ) = 0. 4

(7.5)

(7.6)

Now, make the following transformation of dependent variables [u(t)]2 = |x(t)|q , 2

(7.7a)

p

[v(t)] = |y(t)| ,

(7.7b)

and note that Equation (7.1) can be written as [v(t)]2 + [u(t)]2 = 1,

(7.8)

in the new variables. The work presented in Chapter 5 informs us that there are an unlimited number of periodic solutions to Equation (7.8) provided that u(t) and v(t) take the form u(t) = cos ψ(t), where

(

v(t) = sin ψ(t),

(7.9)

ψ(−t) = −ψ(t), ψ(t + T ) = ψ(t) + 2π,

(7.10)

and the general solution for ψ(t) is ψ(t) =

∞ X



ak sin

k=1

2πkt T



+



2π T



t.

(7.11)

As a consequence of these results, x(t) and y(t) can be selected as follows x(t) = [cos ψ(t)]2/q

sgn(cos ψ(t)),

(7.12a)

2/p

sgn(sin ψ(t)).

(7.12b)

y(t) = [sin ψ(t)] Comments:

(a) Observe that in Equation (7.1), the absolute value operation appears. This takes care of potential problems of having the sign functions appearing in Equation (7.12).

GENERALIZED TRIGONOMETRIC FUNCTIONS: |y|p + |x|q = 1

111

(b) When actually computing either x(t) and y(t), from the expressions given in Equations (7.12), the following mathematical formats must be used  1/q [cos ψ(t)]2/q → [cos ψ(t)]2 , (7.13a)  2/p 2 1/q [sin ψ(t)] → [sin ψ(t)] . (7.13b) Doing this prevents the appearance of complex values at an intermediate step of the calculations.

7.3

SUMMARY

For purposes of convenience, we give below a concise summary of the results derived in Section 7.2. Also, hopefully, without generating confusion, we rewrite and renumber several previously given equations. The functional equation ( |y|p + |x|q = 1, (7.14) p > 0, q > 0, has the following class of periodic solutions xpq (t, T ) = [cos ψ(t)]2/q

sgn(cos ψ(t)),

(7.15)

2/p

sgn(sin ψ(t)),

(7.16)

ypq (t, T ) = [sin ψ(t)]

where ψ(t) has a Fourier sine series as given by Equation (7.11). Note that in Equations (7.15) and (7.16), the dependence on (p, q, T ) is indicated explicitly. It is important to note that xpq (t, T ) and ypq (t, T ) were explicitly constructed to mimic the trigonometric cosine and sine functions. However, unlike most of the previous work on generalized trigonometric cosine and sine functions, our method of construction is not based on an a priori inverse integral relation.

7.4

GALLERY OF PARTICULAR SOLUTIONS

We end this chapter with a gallery of solutions for our class of periodic functions. On the following pages, we give plots of |y|p + |x|q = 1, and xpq (t) and ypq (t) versus t, for selected (p, q) values.

112 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS

1.5

y

1.0

0.5

- 1.0

- 0.5

O

0.5

1.0

- 0.5

- 1.0

- 1.5

FIGURE 7.4.1a: Plot of |y|1.2 + |x|2 = 1, i.e., p = 1.2 and q = 2.

x

GENERALIZED TRIGONOMETRIC FUNCTIONS: |y|p + |x|q = 1

x

1.0

0.5

5

10

15

20

25

5

10

15

20

25

- 0.5

- 1.0

y

1.0

0.5

- 0.5

- 1.0

FIGURE 7.4.1b: Plots of x1.2,2 (t) and y1.2,2 (t), versus t.

113

114 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS

1.5

y

1.0

0.5

- 1.0

O

- 0.5

0.5

1.0

- 0.5

- 1.0

- 1.5

FIGURE 7.4.2a: Plot of |y|2 + |x|4/3 = 1, i.e., p = 2 and q = 4/3.

x

GENERALIZED TRIGONOMETRIC FUNCTIONS: |y|p + |x|q = 1

1.0

0.5

5

10

15

20

25

5

10

15

20

25

- 0.5

- 1.0

y

1.0

0.5

- 0.5

- 1.0

FIGURE 7.4.2b: Plot of x2,4/3 (t) and y2,4/3 (t), versus t.

115

116 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS

1.5

y

1.0

0.5

- 1.0

- 0.5

O

0.5

1.0

- 0.5

- 1.0

- 1.5

FIGURE 7.4.3a: Plot of |y|3 + |x|4 = 1, i.e., p = 3 and q = 4.

x

GENERALIZED TRIGONOMETRIC FUNCTIONS: |y|p + |x|q = 1

x

1.0

0.5

5

10

15

20

25

5

10

15

20

25

- 0.5

- 1.0

y

1.0

0.5

- 0.5

- 1.0

FIGURE 7.4.3b: Plot of x3,4 (t) and y3,4 (t), versus t.

117

Chapter 8 GENERALIZED TRIGONOMETRIC HYPERBOLIC FUNCTIONS: |y|p − |x|q = 1

8.1

INTRODUCTION

We will use the functional equation |y|p − |x|q = 1

(8.1)

to construct generalized hyperbolic functions, in correspondence with the standard trigonometric hyperbolic functions, cosh(at) and sinh(at), eat + e−at , 2 eat − e−at . hyperbolic sine(at) ≡ sinh(at) = 2 These functions satisfy the functional relation hyperbolic cosine(at) ≡ cosh(at) =

[cosh(at)]2 − [sinh(at)]2 = 1,

(8.2) (8.3)

(8.4)

which is a special case of Equation (8.1) with p = 2 and q = 2. Note that the cosh and sinh functions have the following properties: (i) cosh(at) and sinh(at) are, respectively, even and odd, i.e., cosh(−at) = cosh(at),

sinh(−at) = − sinh(at).

(8.5)

(ii) Since cosh and sinh are continuous functions, we have (as a trivial consequence) cosh(0) = 1, sinh(0) = 1. (8.6) (iii) The following asymptotic relations hold (for a > 0) (
cosh(at) ∼ 21 eat ,
t → +∞ : sinh(at) ∼ 12 eat , (
cosh(at) ∼ 12 e−at ,
t → −∞ : sinh(at) ∼ (−) 21 e−at .

(8.7)

(8.8)

119

120 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS

8.2

SOLUTIONS

Denote by ypq (t) and xpq (t) the generalized hyperbolic cosine and sine solutions of Equation (8.1). We wish these functions to have the following properties: (a) ypq (0) = 1,

xpq (0) = 0.

(8.9)

(b) ypq (t) = 1 and xpq (t) = 0 are to be, respectively, even and odd functions of t, i.e., ypq (−t) = ypq (t), xpq (−t) = −xpq (t). (8.10) Carrying out a similar set of substitutions and related procedures as was done in Chapter 7, the following expressions are gotten for ypq (t) and xpq (t): ypq (t) = [cosh(at)]2/p ,

(8.11)

 1/q xpq (t) = [sinh(at)]2 [sgn(t)],

(8.12)

where a is taken to be positive. Note that the expression, enclosed in {. . . }, for xpq (t) is written in a manner such that xpq (t) is always real. Inspection of Equations (8.11) and (8.12) show that they satisfy the a priori desired properties expressed in Equations (8.9) and (8.10). These functions also have the following asymptotic behaviors (
2/p 2at ypq (t) ∼ 21 e p , t → +∞ : (8.13)
1 2/q 2at xpq (t) ∼ 2 e q , t → (−)∞ :

8.3

(

ypq (t) ∼

xpq (t) ∼


1 2/p − 2at e p , 2
2/q − 2at (−) 12 e q .

(8.14)

GALLERY OF SPECIAL SOLUTIONS

We now present a gallery of ypq and xpq for the same values of (p, q) as presented in Section 7.4 of the previous chapter. Observe that from a “pictorial” viewpoint, the graphs are similar in shape. All plots are for a = 1.

GENERALIZED TRIG HYPERBOLIC FUNCTIONS: |y|p − |x|q = 1 121

y 10

5

-4

-2

2

4

-5

- 10

FIGURE 8.3.1a: Plot of |y|1.2 − |x|2 = 1, for p = 1.2 and q = 2.

x

122 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS

y 10

5

-3

-2

-1

1

2

-5

- 10

10 5

-2

-1

1

2

-5 - 10

FIGURE 8.3.1b: Plots of y1.2,2 (t) and x1.2,2 (t), versus t.

3

GENERALIZED TRIG HYPERBOLIC FUNCTIONS: |y|p − |x|q = 1 123

y 10

5

- 10

-5

5

10

-5

- 10

FIGURE 8.3.2a: Plot of |y|2 − |x|4/3 = 1, for p = 2 and q = 4/3.

x

124 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS

y 10

5

-2

-1

1

2

-5

- 10

10 5

-3

-2

-1

1

2

3

-5 - 10

FIGURE 8.3.2b: Plots of y2,4/3 (t) and x2,4/3 (t), versus t.

GENERALIZED TRIG HYPERBOLIC FUNCTIONS: |y|p − |x|q = 1 125

y 10

5

-4

-2

2

4

-5

- 10

FIGURE 8.3.3a: Plot of |y|3 − |x|4 = 1, for p = 3 and q = 4.

x

126 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS

y 10

5

-4

-2

2

4

-5

- 10

10 5

-4

-2

2

4

-5 - 10 FIGURE 8.3.3b: Plots of y3,4 (t) and x3,4 (t), versus t.

Chapter 9 APPLICATIONS AND ADVANCED TOPICS

9.1

INTRODUCTION

This chapter examines a number of issues related to conservative dynamical systems for which the Hamiltonian takes the form  p q  H(x, y) = |y| + |x| = 1 (9.1) p ≥ 1, q ≥ 1   x(0) = 1, y(0) = 0,

where x = x(t) and y(y(t). The independent variable t is the time. The corresponding equations of motion are ∂H(x, y) dx = , dt ∂y

dy ∂H(x, y) =− , dt ∂x

(9.2)

and dx = p|y|p−1 sgn(y), dt dy = −q|x|q−1 sgn(x), dt

x(0) = 1,

(9.3a)

y(0) = 0.

(9.3b)

Observe that for p = q = 2, Equations (9.1) and (9.3) take the form ( H(x, y) = y 2 + x2 = 1, (9.4) dy dx dt = y, dt = −x, while for p = q = 1, we have ( H(x, y) = |y| + |x| = 1, dy dx dt = sgn(y), dt = −sgn(x).

(9.5)

Note that Equations (9.3) are invariant under the dependent variables transformation ( x(t) → −x(t), (9.6) y(t) → −y(t). 127

128 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS We denote such systems as having odd parity, and call them odd-parity systems. For the general case of a second-order differential equation, which we write as F (x, dx/dt, d2 x/dt2 ) = 0, (9.7) it is of odd parity if and only if F (−x, −dx/dt, −d2 x/dt2 ) = (±)F (x, dx/dt, d2 x/dt2 ).

(9.8)

The following are examples of odd-parity systems defined in terms of differential equations d2 x + x = 0, dt2 d2 x Duffing oscillator : 2 + a1 x + b1 x3 = 0, dt d2 x dx van der Pol oscillator : 2 + x = ǫ(1 − x2 ) , dt dt harmonic oscillator :

(9.9) (9.10) (9.11)

where (a, b, ǫ) are parameters. In Section 9.3, we show that for odd-parity periodic systems, the Fourier expansions have only odd harmonics. For many dynamical systems, modeling a broad range of phenomena in the natural and engineering sciences, Equation (9.7) takes the special form   d2 x dx + f x, = 0, (9.12) dt2 dt and the system is of odd parity if and only if     dx dx f −x, − = −f x, . dt dt

(9.13)

See the expressions in Equations (9.9), (9.10), and (9.11). We now present arguments to indicate why the harmonic oscillator, Equation (9.9), and the Duffing oscillator, Equation (9.10), appear so often as mathematical models of dynamical systems. To begin, assume that x is the extension of a one-dimensional system away from a position of equilibrium. Further, assume that based on Newton’s force law, we have d2 x = −f (x). dt2

(9.14)

Expanding f (x) in a Taylor series gives d2 x = f0 + f1 x + f2 x2 + f3 x3 + O(x4 ) dt2 where fk =



1 k!



dk f (0) , dxk

k = (0, 1, 2, . . . ).

(9.15)

(9.16)

APPLICATIONS AND ADVANCED TOPICS

129

Now f0 is zero, since our reference point is an equilibrium point. Also, for f2 = 0, the system is of odd parity. Therefore, neglecting terms of O(x4 ) and higher, we obtain d2 x + f1 x + f3 x3 = 0, (9.17) dt2 which is the Duffing equation. Note that the harmonic oscillator equation is the special case for when f3 = 0. If y is taken to be y = dx/dt, then in the x-y phase-plane d2 x dy(x) = y(x) , 2 dt dx

(9.18)

and Equation (9.17) can be written as y

dy + f1 x + f3 x3 = 0. dx

Integrating this expression gives       1 f1 f3 2 2 ¯ H(x, y) = y + x + x4 = constant, 2 2 4

(9.19)

(9.20)

which is the energy function or Hamiltonian of the system. Therefore, the equations of motion are  ¯  dx = ∂ H(x,y) = y, dt ∂y (9.21) ¯  dy = − ∂ H(x,y) = −f1 x − f3 x3 , dt

∂x

which is equivalent to the second-order differential Equation (9.17). ¯ Observe that if f1 = 0, then H(x, y) becomes     1 f3 2 ¯ H(x, y) = y + x4 = constant 2 4

(9.22)

and with a rescaling of the x and y variables, this becomes Equation (9.1) with p = 2 and q = 4. For the remainder of this chapter, we will investigate dynamical systems for which the Hamiltonian has only two terms, both of which are power laws of their respective variables. Thus, we directly demonstrate the importance of generalized, periodic, trigonometric functions for the modeling, analysis, and understanding of this class of dynamical systems. In Section 9.2, we give arguments to show that odd-parity periodic systems have Fourier expansions for which only the odd harmonics or frequencies appear. Section 9.3 introduces the concept of a “truly nonlinear oscillator (TNLO)” and we examine and solve special cases of this type of oscillator. In Section 9.4, the so-called Ateb periodic functions are introduced and briefly

130 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS discussed. Discretization of differential equations plays an important role in the numerical integration of such equations. Section 9.5 presents exact discretizations of the differential Equation (9.17) for which the corresponding solutions are the three Jacobi elliptic functions. The next two Sections, 9.6 and 9.7, are concerned with the method of harmonic balance and its generalization. This technique is of great value when accurate analytical approximations to periodic solutions are required. Another methodology created to calculate approximations to the periodic solutions of the equations of motion, see Equations (9.3), is based on iteration procedures. In Section 9.8, we outline these types of procedures and illustrate their use by applying these methods to several examples. Finally, the main reason for the necessity of this chapter is the fact that in general, exact analytical solutions to Equation (9.1), in terms of finite combinations of “elementary functions,” are not possible. Therefore, we must either obtain numerical representations of the solutions and/or construct analytical approximations. The latter issue is the focus of this chapter.

9.2

ODD-PARITY SYSTEMS AND THEIR FOURIER REPRESENTATIONS

In this section, arguments are presented to show that for odd-parity systems the Fourier series for periodic solutions only contain odd (angular) frequencies. To begin, consider the following dimensionless Duffing equation ( 2 d x 3 dt2 + x + ǫx = 0, (9.23) dx(0) x(0) = 1, dt = 0. For 0 < ǫ ≪ 1, the corresponding perturbation solution is  ǫ  x(θ, ǫ) = cos θ + (− cos θ + cos 3θ) 32  2  ǫ + (23 cos θ − 24 cos 3θ + cos 5θ) 1024 + O(ǫ3 ),

(9.24a)

and  3ǫ 21ǫ2 3 θ(ǫ, t) ≡ ω(ǫ)t = 1 + − + O(ǫ ) , 8 256 

(9.24b)

where the angular frequency, ω, is related to the period, T , by the relation ω=

2π . T

(9.25)

APPLICATIONS AND ADVANCED TOPICS

131

Observe that, at least to terms of O(ǫ3 ), only odd angular frequencies occur. Likewise, for the van der Pol oscillator
dx d2 x + x = ǫ 1 − x2 , dt2 dt

(9.26)

which is of odd parity, the perturbation solution for its unique limit-cycle solution is ǫ x(θ, ǫ) = 2 cos θ + (3 sin θ − sin 3θ)  2 4 ǫ (−13 cos θ + 18 cos 3θ − 5 cos 5θ) + 96 + O(ǫ3 ),

(9.27a)

where  ǫ2 3 θ(ǫ, t) ≡ ω(ǫ)t = 1 − + O(ǫ ) t. 16 

(9.27b)

Again, only odd angular frequencies appear in the solution. We now demonstrate that this is a general feature of odd-parity periodic systems, i.e., if x(t) is such a solution, then x(t) has a Fourier representation x(t) =

∞ X

[Ak cos(2k − 1)ωt + Bk sin(2k − 1)ωt].

(9.28)

k=1

The demonstration is based on the following three requirements: (i) The equation of motion d2 x +F dt2



dx x, dt



= 0,

(9.29)

is of odd parity and has a periodic solution. (ii) In the two-dimensional phase-space, (x, y = dx/dt), the periodic solutions occur about the fixed point, (0, 0). (iii) The periodic solutions of Equation (9.29) are essentially unique, i.e., if x = φ(t) is a non-trivial periodic solution, then for t0 6= 0, z = φ(t − t0 ), is also a periodic solution. From the perspective of the phase-space, the moving point   dφ(t) (x(t), y(t)) = φ(t), dt

(9.30)

132 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS traces out a closed curve. This means that     dz(t) dφ(t − t0 ) z(t), = φ(t − t0 ), , dt dt

(9.31)

traces out the same closed curve, except that it is shifted in phase. Now assume Equation (9.29) has the complex Fourier representation x(t) =

∞ X

[ak eikωt + ak e−ikωt ],

(9.32)

k=1

where the ak are complex valued coefficients. Note that if x(t) is a periodic solution, then z(t) is also a periodic solution, where z(t) is defined to be   T z(t) = −x t + . (9.33) 2 This result is a consequence of the fact that both x(t) and −x(t) are solutions, and thus x(t − t0 ) and −x(t − t0 ) are also periodic solutions. From the uniqueness requirement z(t) = x(t) (9.34) and

  T = −x(t). x t+ 2

(9.35)

If Equation (9.32) is substituted into this last expression and a comparison is made of the coefficients of the two exponential terms, then the following result is found (−1)k ak = −ak , (9.36) and this implies that all the even labeled coefficients are zero, a2m = 0,

m = (1, 2, 3 . . . ).

(9.37)

Therefore, x(t) has a Fourier representation only in terms of odd harmonics or angular frequencies; see Equation (9.28). This result will play an important role in the formulation of appropriate methods to determine approximate analytical solutions for odd-parity systems. A final comment. Consider the function N (x), defined as ( 2n+1 N (x) = x 2m+1 , (9.38) n = 0, 1, 2, . . . ; m = 0, 1, 2, . . . . A direct calculation shows that N (x) is an odd function, i.e., N (−x) = −N (x),

(9.39)

APPLICATIONS AND ADVANCED TOPICS

133

and, as a consequence, the following differential equations are of odd parity: d2 x + x1/3 = 0, dt2  1/3 dx d2 x 5/3 , +x = ǫ(1 − |x|) dt2 dt

d2 x x7/3 + = 0. dt2 1 + x2

9.3

(9.40a) (9.40b) (9.40c)

TRULY NONLINEAR OSCILLATORS

The dynamic systems modeled by Hamiltonians having the structure given in Equation (9.1) represent “truly nonlinear (TNL)” oscillators. To understand what these oscillators are, we begin with the following definition. Definition 9.3.1 A function f (x) is a TNL function, at x = 0, if and only if f (x) has no linear approximation in any neighborhood of x = 0. Explicit examples of TNL functions are ( f1 (x) = x1/3 , f2 (x) = x3 , f3 (x) = |x|, x3 f4 (x) = x + x1/3 , f5 (x) = 1+x 2.

(9.41)

Definition 9.3.2 If f (x) is a TNL function and if solutions to the differential equation d2 x + f (x) = 0, (9.42) dt2 are periodic, then Equation (9.42) is a TNL oscillator. Note that if H(x, y), in Equation (9.1), is taken as the Hamiltonian of a system, then its “solutions,” x = x(t) and y = y(t), are periodic, and the corresponding equations of motion, Equations (9.3), are a coupled set of nonlinear, first-order, ordinary differential equations. Using the transformation (x, y) → (−x, −y),

(9.43)

in Equations (9.3), it is easy to see that these equations are of odd parity. For the remainder of this section, we will investigate exact solutions to several cases of Equation (9.1) for which p = 2, i.e., H(x, y) = y 2 + |x|q = 1,

x(0) = 1, y(0) = 0.

(9.44)

The selected q-values correspond to dynamical systems of importance to various phenomena in the natural and engineering sciences.

134 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS

9.3.1

Antisymmetric, Constant Force Oscillator

The relevant equation of motion for this case is d2 x + sgn(x) = 0 dt2

(9.45)

where, as a reminder,   +1, x > 0, sgn(x) = 0, x = 0,   −1, x < 0.

(9.46)

Consequently, Equation (9.45) is equivalent to the following two linear differential equations d2 x + 1 = 0, dt2 d2 x − 1 = 0, dt2

x > 0,

(9.47a)

x < 0,

(9.47b)

which have the respective solutions   1 2 (+) t + A1 t + B1 , x (t) = − 2   1 2 t + A2 t + B2 , x(−) (t) = 2

(9.48a) (9.48b)

where (A1 , A2 , B1 , B2 ) are integration constants. To construct a solution, start with the initial condition x(0) = 0,

dx(0) = A > 0. dt

(9.49)

Since A > 0, we must use x(+) (t) and impose the initial conditions given in Equation (9.49); doing this gives ( x(+) (0) = B1 = 0, (9.50) dx(+) (0) = A1 = A, dt and, therefore, (+)

x

  1 (t) = − t(t − 2A), 2

0 ≤ t ≤ 2A.

(9.51)

Inspection of Equation (9.51) shows that x(+) (t) has the following features:  (+)  x (0) = 0, (+) (9.52) x (t) > 0, 0 < t < 2A,   (+) x (2A) = 0.

APPLICATIONS AND ADVANCED TOPICS

135

These results clearly imply that the period, T , of the oscillation, is T = 4A.

(9.53)

The conditions on x(t) at t = 2A and t = 4A are x(−) (2A) = 0,

x(−) (4A) = 0,

(9.54)

B2 = 4A2 ,

(9.55)

and from Equation (9.48b), we find A2 = −3A, and x(−) (t) is x(−) (t) =

  1 2 t − (3A)t + 4A2 , 2

2A ≤ t ≤ 4A.

(9.56)

In summary, the periodic solution to Equation (9.45), with the initial conditions in Equation (9.49), is (
0 ≤ t ≤ 2A, − 21 t(t − 2A), (9.57) x(t) =
2 1 2 2 t − (3A)t + 4A , 2A ≤ t ≤ 4A,

with the periodicity condition

x(t + 4A) = x(t).

(9.58)

From this piecewise continuous representation for x(t), its Fourier series can be easily calculated and is given by the expression   ∞   (2k + 1)πt 1 16A2 X . (9.59) sin x(t) = π3 (2k + 1)3 2A k=0

Observe first that only odd angular frequencies occur in the expansion, i.e.,   2π ωk = (2k + 1) = (2k + 1)ω0 , (9.60) 4A where the period is T = 4A. Second, x(t) is an odd function of t. Third, the Fourier coefficients, bk , where   1 16A2 , (9.61) bk = π3 (2k + 1)3 decrease asymptotically as k −3 . Finally, this system corresponds to the following H(x, y)   1 y 2 + |x|, H(x, p) = 2

(9.62)

which on comparison with Equation (9.1) and rescaling the y-variable give p = 2,

q = 1.

(9.63)

136 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS

9.3.2

Particle in a Box

Many physical systems can be approximated as taking place in a onedimensional box. To begin, assume that the box is located between x = 0 and x = L. Thus, the size of the box is L. Assume that a (classical) particle is located in the box and that at t = 0, it is at x = 0, with velocity v0 , i.e., it is moving to the right. After a time T ∗ , where T∗ =

L , v0

(9.64)

the particle hits the wall at x = L, and reverses direction. Then, after a time T ∗ , it hits the wall at x = 0, and reverses direction. Assuming all the collisions are elastic, the overall motion is periodic with period T , where T = 2T ∗ =

2L . v0

(9.65)

In terms of x(t), this feature is given by the periodicity condition x(t + T ) = x(t)

(9.66)

with a related expression for the velocity v(t) = dx(t)/dt, i.e., v(t + T ) = v(t).

(9.67)

The time dependencies of x(t) and v(t) are ( v0 t, 0 ≤ t ≤ T2 , x(t) = v0 (T − t), T2 ≤ t ≤ T , and v(t) =

(

v0 , 0 ≤ t ≤ T2 , −v0 , T2 ≤ t ≤ T .

(9.68)

(9.69)

A rescaling of the x(t) and v(t) variables to units such that v0 = 1 and L = π, simplifies the Fourier series representations for x(t) and v(t). With these units, we have the (rescaled) period, T = 2π, and the expressions  X ∞ 4 cos(2k − 1)t π , (9.70a) x(t) = − 2 π (2k − 1)2 k=1  X ∞ sin(2k − 1)t 4 . (9.70b) v(t) = π (2k − 1) k=1

Observe the following:

(i) The Fourier series contains only odd harmonics. (ii) Equation (9.70a) gives the following well-known expression for π 2 π2 = 8

∞ X

k=1

∞

X 1 1 =8 . 2 (2k − 1) (2k + 1)2 k=0

(9.71)

APPLICATIONS AND ADVANCED TOPICS

9.3.3

137

Restricted Duffing Equation

The restricted Duffing equation d2 x + x3 = 0, dt2 is a special case of the general Duffing equation

(9.72)

d2 x dx +a + bx + cx2 + dx3 = 0, dt2 dt

(9.73)

a = 0,

(9.74)

where b = 0,

c = 0,

d = 1.

For the initial conditions dx(0) = 0, (9.75) dt the exact solution√can be expressed in terms of the Jacobi cosine elliptic function, cn(At; 1/ 2), i.e., √ x(t) = A cn(At; 1/ 2). (9.76) x(0) = A,

The Fourier series for “cn” is rather complex. Let (k, k ′ ) satisfy the relation

and define q(k) as

(k)2 + (k ′ )2 = 1,

(9.77)

   πF (k ′ ) , q(k) = exp − F (k)

(9.78)

where F (k) is the complete elliptical integral of the first kind, i.e., Z π/2 dψ p F (k) = . 1 − k 2 sin2 ψ 0

(9.79)

With these definitions, the general Jacobi elliptic function has the Fourier series representation ! X  1 ∞ q m+ 2 2π cos(2m + 1)v, (9.80) cn(u, k) = kF (k) m=0 1 + q 2m+1 where

 π u. 2F (k) For the restricted Duffing equation, we have    k = √1 , F √1 = 1.854 074 . . . , 2  2  q √12 = 0.043 213 . . . . v = v(k, u) =



(9.81)

(9.82)

Using the results given in Equations (9.77) to (9.82), the Fourier series can be constructed. Again, note that only odd harmonics appear.

138 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS

9.4

ATEB PERIODIC FUNCTIONS

Consider the following coupled, first-order, nonlinear differential equations   dy(t) dx(t) 2 y(t), =− = x(t)α (9.83) dt α+1 dt with initial conditions x(0) = 1,

y(0) = 0.

(9.84)

In the x-y phase-plane, there is a fixed point at (¯ x, y¯) = (0, 0). Also, x(t) satisfies the following second-order differential equation   d2 x(t) 2 x(t)α = 0. (9.85) + dt2 α+1 If this equation is to be of odd parity, then α must either have the form ( 2n+1 α = 2m+1 , (9.86) n = 0, 1, 2, 3, . . . ; m = 0, 1, 2, 3, . . . , or the term x(t)α replaced by xα → |x|α sgn(x).

(9.87)

The trajectories of the curves in the x-y phase-space are solutions to the first-order differential equation  α  2n + 1 x dy α+1 , α= =− (9.88a) dx 2 y 2m + 1 or dy =− dx



α+1 2



|x|α sgn(x) y



,

α = real > 0.

(9.88b)

A first integral can be found by rewriting Equation (9.88) to the forms (
α 2n+1 y dy + α+1 , x dx = 0, α = 2m+1 2 (9.89)
α+1 α y dy + 2 |x| sgn(x)dx = 0, α = real > 0,

and integrating to obtain ( y2 xα+1 = constant, 2 + 2
α+1 y2 1 = constant, 2 + 2 |x|

α=

2n+1 2m+1 ,

α = real > 0.

(9.90)

APPLICATIONS AND ADVANCED TOPICS

139

Imposing the initial conditions, x(0) = 1 and y(0) = 0, gives ( y2 xα+1 2n+1 = 21 , α = 2m+1 , 2 + 2
2 y 1 1 α+1 = 2 , α = real > 0. 2 + 2 |x|

(9.91)

Comparing these expressions with Equation (9.1) gives p = 2 and q = α + 1. In an important paper by Rosenberg, published in 1963, he associated the name “Ateb functions” with x(t) and y(t), and examined the most important of their mathematical properties. In our notation, we will call x(t) and y(t), respectively, the Ateb-cosine and Ateb-sine functions, i.e., ( Atc(α, t) ≡ x(t), (9.92) Ats(α, t) ≡ y(t). Let Π(α) be defined as Π(α) ≡ B



1 1 , α+1 2



,

(9.93)

where B(a, b) is the beta function, i.e., B(a, b) =

Γ(a)Γ(b) , Γ(a + b)

(9.94)

then the following properties can be easily established: (i) Atc(t) and Ats(t) are, respectively, even and odd, i.e., Atc(−t) = Atc(t),

Ats(−t) = −Ats(t).

(ii) Both Atc(t) and Ats(t) are periodic, with period 2Π(α), i.e., ( Atc(t + 2Π(α)) = Atc(t), Ats(t + 2Π(α)) = Atc(t).

(9.95)

(9.96)

(iii) Atc(t) and Ats(t) satisfy the following coupled, first-order pair of differential equations   2 d Ats(t), (9.97a) Atc(t) = − dt α+1 d Ats(t) = (Atc(t))α . (9.97b) dt (iv) Atc(t) = x(t) is a solution to the following nonlinear, second-order differential equation   d2 x 2n + 1 2 xα = 0, α= + , (9.98a) 2 dt α+1 2m + 1   2 d2 x |x|α sgn(x) = 0, α = real > 0. (9.98b) + dt2 α+1

140 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS It should be indicated that the periodic Ateb functions have found a significant use in the analysis of certain nonlinear, conservative oscillatory systems.

9.5

EXACT DISCRETIZATION OF THE JACOBI ELLIPTIC DIFFERENTIAL EQUATIONS

The Duffing differential equation provides a nontrivial mathematical model for a broad range of phenomena in the natural and engineering sciences. It also corresponds to the simplest generalization of the so-called simple harmonic oscillator equation d2 y + y = 0. (9.99) dt2 For our purposes, we consider the Duffing equation in the form d2 y + ay + by 3 = 0, dt2

(9.100)

where (a, b) are real parameters. The following initial conditions and parameter signs give solutions that are, respectively, related to the Jacobi cosine and sine elliptic functions: dy(0) = 0; a > 0, b > 0; y(t) = Acn(t) (9.101a) dt dy(0) = A; a > 0, b < 0; y(t) = Asn(t). (9.101b) y(0) = 0, dt The main goal of this section is to construct discretizations of the Duffing differential equation such that at the discrete-time lattice points, the solutions of the difference equations are exactly equal to the solutions of the differential equations. This effort can play an important role in the numerical integration of the equations of motion for dynamical systems modeled by the Duffing equation. y(0) = A,

9.5.1

Rescaled Duffing Equation

The Duffing differential equation can be rescaled using the substitutions y(t) = Ax(t¯),

t = T t¯,

(9.102)

where (A, T ) are the scaling parameters, and x and t¯ are the new dependent and independent variables. Substitution of this result into Equation (9.100) gives d2 x + (aT 2 )x + (bT 2 A2 )x3 = 0. (9.103) dt¯2 In the next sections, we show how to select A and T .

APPLICATIONS AND ADVANCED TOPICS

9.5.2

141

Exact Difference Equation for CN

We now determine the exact difference equation for the Jacobi cosine function cn(t, k). In the calculations to follow, the parameter k will not be indicated. To begin, we make use of an addition theorem for cn(t). cn(u + v) = Now define

(

(cn u)(cn v) − (sn u)(dn u)(sn v)(dn v) . 1 − k 2 (sn u)2 (sn v)2

t¯N = hN,

h = ∆t, xN = cn(t¯N ), N = 0, 1, 2, . . . ,

(9.104)

(9.105)

and use Equation (9.104) to obtain xN +1 + xN −1 =

1−

2(cn h)xN 2 k (sn h)2 (1 −

x2N )

.

(9.106)

To obtain this latter result, we made use of the relation [sn(tN )]2 = 1 − [cn(tN )]2 = 1 − x2N .

(9.107)

Adding (−2xN ) to both sides of Equation (9.106), and replacing (cn h)xN by (cn h)xN = xN − [1 − (cn h)]xN ,

(9.108)

we obtain 

   1 − (cn h) xN +1 + xN −1 2 x − k N (sn h)2 2   xN +1 + xN −1 + 2k 2 x2N = 0. (9.109) 2

xN +1 − 2xN + xN −1 +2 (sn h)2

This second-order, nonlinear, difference equation has xN = cn(tN ) = cn(tN , k)

(9.110)

as its solution. The differential equation corresponding to this difference equation is determined by taking the following limits h → 0,

N → ∞,

t¯ = hN = fixed.

(9.111)

Using also the results (

sn h = h + O(h2 ) 2 cn h = 1 − h2 + O(h4 ),

(9.112)

142 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS Equation (9.109) becomes in these limits the differential equation d2 x + (1 − 2k 2 )x + 2k 2 x3 = 0. dt¯2

(9.113)

Comparing this with Equation (9.103) and equating the similar coefficients of x and x3 , gives the relations 1 − 2k 2 = aT 2 ,

2k 2 = bT 2 A2 .

(9.114)

If these equations are now solved for T 2 and k 2 , we obtain T2 =

1 , a + bA2

k2 =

bA2 , 2(a + bA2 )

(9.115)

where from the second expression, we find that k 2 satisfies the bounds 0 < k2
0 and b < 0, we have T2 =

2 , 2a − |b|A2

k2 =

|b|A2 . 2a − |b|A2

(9.122)

Since we must have 0 < k 2 < 1, it follows that the amplitude is restricted in magnitude, i.e., r a . (9.123) |A| < |b|

9.6

HARMONIC BALANCE: DIRECT METHOD

9.6.1

Methodology

Harmonic balance procedures are a set of techniques which allow the calculation of analytical approximations to the periodic solutions of differential equations. These methods are based on representing a given periodic solution by means of a truncated Fourier series, i.e., a trigonometric polynomial. We now outline this procedure. For our purposes, assume that the differential equation under consideration can be written in the form   dx d2 x = 0. (9.124) , F x, dt dt2 Further assume that it is of odd parity, i.e.,     dx d2 x dx d2 x F −x, − , − 2 = −F x, , 2 dt dt dt dt

(9.125)

and it has a periodic solution which can be expressed as a Fourier series, i.e., x(t) =

∞ X

k=1

{Ak cos[(2k − 1)Ωt] + Bk sin[(2k − 1)Ωt]]} .

(9.126)

Let xN (t) be the N -th-order harmonic balance approximation to x(t). It corresponds to the representation xN (t) =

N X  N ¯Nt + B ¯ N sin(2k − 1)Ω ¯ N (t) , A¯k cos(2k − 1)Ω k

k=1

(9.127)

144 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS ¯N ¯ where (A¯N k , Bk , ΩN ) are approximations to (Ak , Bk , Ω) for k = 1, 2, . . . , N . In general, for conservative, one-space-dimensional systems, Equation (9.124) takes the form d2 x = f (x), (9.128) dt2 and the initial conditions can be selected to be x(0) = A,

dx(0) = 0, dt

(9.129)

with the consequence that only cosine terms appear in the Fourier expansion, i.e., ∞ X x(t) = Ak cos(2k − 1)Ωt, (9.130) k=1

and

xN (t) =

N X

k=1

¯ A¯N k cos(2k − 1)ΩN t.

(9.131)

In the work to follow, only conservative dynamical systems will be discussed. From Equation (9.1), this situation corresponds to p = 2. ¯N ¯N Note that xN (t) has (N +1) unknowns: the N -coefficients, (A¯N 1 , A2 , . . . , AN ) ¯ and ΩN . These quantities can be determined by performing the following steps: (i) Substitute Equation (9.131) into Equation (9.128) and expand into harmonics, such that it has the form N X

k=1

¯ N t] + HOH ≈ 0. Hk cos[(2k − 1)Ω

(9.132)

¯ N , i.e., The Hk will be functions of the N -coefficients and Ω
¯N ¯N ¯ Hk ≡ Hk A¯N 1 , A2 , . . . , AN , ΩN .

(9.133)

Hk = 0,

(9.134)

(ii) Now equate the N -functions to zero, i.e.,

k = 1, 2, . . . , N.

¯N ¯N ¯ ¯N (iii) Solve these N -equations for A¯N 2 , A3 , . . . AN , and ΩN , in terms of A1 . N ¯ (iv) Use the initial conditions, see Equation (9.129), to determine A1 , i.e., xN (0) = A = A¯N 1 +

N X

¯N A¯N k (A1 ).

(9.135)

k=2

We now illustrate the use of the harmonic balance method by applying it to

APPLICATIONS AND ADVANCED TOPICS

145

two systems for which their Hamiltonians have the structure given in Equation (9.1). For all examples, we have p = 2. We will also use the following compact notation for time derivatives x(t) ˙ =

dx(t) , dt

x ¨(t) =

d2 x(t) . dt2

(9.136)

Finally, it should be pointed out that the same initial conditions are imposed on each xN (t), i.e., xN (0) = A and x˙ N (0) = 0.

9.6.2

x ¨ + x3 = 0

The equation of motion x ¨ + x3 = 0,

x(0) = 1,

x(0) ˙ = 0,

has the Hamiltonian function (

H(x, y) = 21 y 2 + 41 x4 = x(0) = 1, y(0) = 0,

1 4

(9.137)

(9.138)

where y(t) = dx(t)/dt = x(t). ˙ Note that with a rescaling of the variables, H(x, y) corresponds to Equation (9.1) with p = 2 and q = 4. However, for our calculations, we will keep the results in Equation (9.138). The N = 1 approximation, x1 (t), is x1 (t) = A cos Ω1 t,

(9.139)

where, from now on, we drop the “bar” on coefficients and the angular frequency Ω. Substituting Equation (9.139) into Equation (9.137) gives (θ = Ω1 t)

and

Therefore,

(−AΩ21 ) cos θ + (A cos θ)3 ≃ 0

(9.140)

    3 A2 cos θ + HOH ≃ 0. A −Ω21 + 4

(9.141)

Ω21 =

  3 A2 , 4

and the first-order harmonic balance solution is # "  1/2 3 At . x1 (t) = A cos 4

(9.142)

(9.143)

With the initial conditions x1 (0) = 1 and x˙ 1 (0) = 0,

(9.144)

146 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS we have

"  # 1/2 3 x1 (t) = cos t . 4

(9.145)

The next approximation, x2 (t), has the form x2 (t) = A1 cos θ + A2 cos 3θ,

θ = Ω2 t.

(9.146)

If this is substituted into Equation (9.137) and the necessary trigonometric expansions are done, then the following result is obtained H1 cos θ + H2 cos 3θ + HOH ≃ 0,

(9.147)

where         3 3 3 H1 = A1 Ω22 − A21 − A1 A2 − A22 , 4 4 2       3 3 1 A21 + A21 A2 + A32 . H2 = −9A2 Ω22 + 4 2 4 If z is defined as z≡

A2 , A1

(9.148) (9.149)

(9.150)

and if H1 is set to zero, we obtain  1/2 3 Ω2 = A1 (1 + z + 2z 2 )1/2 . 4

(9.151)

Now setting H2 to zero and using Ω2 above, we get the following cubic equation for z 51z 3 + 27z 2 + 21z − 1 = 0. (9.152) While this equation has three roots, the one we seek must be real and “small,” i.e., z is the ratio of the amplitudes of the third harmonic, cos(3θ), to that of the first harmonic, cos θ. We expect this ratio to be small if this method is to be valid, i.e., |z| ≪ 1. (9.153) If we denote this root by z1 , then z1 = 0.044818 . . . .

(9.154)

x2 (t) = A1 [cos θ + z1 cos 3θ].

(9.155)

Therefore, x2 (t) is Using the initial condition, x2 (0) = 1, gives 1 = A1 (1 + z1 ),

(9.156)

APPLICATIONS AND ADVANCED TOPICS or

1 , 1 + z1

(9.157)

 1/2   3 (1 + z1 + 2z22 )1/2 = 0.8489. 4 1 + z1

(9.158)

A1 = and Ω2 =

147

The periods T = 2π/Ω, for the first and second approximations, are T1 =

2π = 7.2554, Ω1

T2 =

2π = 7.4016, Ω2

(9.159)

and these are to be compared to the “exact” result Texact = 7.4163.

(9.160)

The corresponding percentage errors, defined as Texact − T · 100, E ≡ percentage error = Texact

(9.161)

are

E1 = 2.2% and E2 = 0.20%.

(9.162)

In summary, the harmonic balance method provides a quick, efficient technique for calculating analytical approximations for the periodic solutions of x ¨ + x3 = 0. In essence, the procedure converts the solving of a strongly nonlinear differential equation to one where nonlinear algebraic equations must be solved. Further application of this technique gives even more accurate results for the period. In addition, the ratio of neighboring amplitude coefficients remains small, a result needed to insure the validity of the method.

9.6.3

x ¨ + x1/3 = 0

The “cube-root” oscillator x ¨ + x1/3 = 0,

x(0) = 1,

x(0) ˙ = 0,

(9.163)

    3 3 1 y2 + x4/3 = , 2 4 4

(9.164)

has the Hamiltonian H(x, y) =

where y(t) = x(t). ˙ The exact angular frequency can be calculated and is found to be Ωexact = 1.070451.

(9.165)

148 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS There are several ways to proceed to determine first- and second-order harmonic balance approximations to the periodic solutions of Equation (9.163). We start with a cubed version of this equation, i.e., (¨ x)3 + x = 0.

(9.166)

x1 (t) = A1 cos(Ω1 t) = A1 cos θ,

(9.167)

For we have on substitution into Equation (9.166)

and

 3 −(Ω1 )2 A1 cos θ + A1 cos θ ≃ 0,     3 A21 (Ω1 )6 cos θ + HOH ≃ 0. A1 1 − 4

(9.168)

(9.169)

This gives for Ω1 the value  1/6 4 Ω1 (A1 ) = 3

1 1/3

A1

!

.

(9.170)

Since x1 (0) = 1, it follows that "  # 1/6 4 A1 = 1 ⇒ x1 (t) = cos t , 3

(9.171)

Ω1 (1) = 1.049115.

(9.172)

and Therefore, the percentage error in the angular frequency, Ω1 , is Ωexact − Ω1 · 100 = 2.0%. E1 = Ωexact If we write x2 (θ) as ( x2 (t) = A1 [cos θ + z cos 3θ], θ = Ω2 t x(0) = 1, x(0) ˙ = 0,

(9.173)

(9.174)

substitute into Equation (9.166), and calculate the coefficients of the cos θ and cos 3θ terms, then the following expressions are obtained        27 243 3 (9.175a) + z+ z 2 = 1, (Ω2 )6 A21 4 4 2        1 27 2187 (Ω2 )6 A21 (9.175b) + z+ z 3 = z. 4 4 4

APPLICATIONS AND ADVANCED TOPICS

149

Dividing these equations gives (1701)z 3 − (27)z 2 + (51)z + 1 = 0,

(9.176)

and the smallest (in magnitude) root is z1 = −0.019178.

(9.177)

Also, from x2 (0) = 1,

1 , 1 + z1 and (leaving out a lot of the details), we calculate A1 =

Ω2 = 1.063410.

(9.178)

(9.179)

The percentage error Ωexact − Ω2 · 100 = 0.7%. E2 = Ωexact

(9.180)

In summary, by rewriting the equation of motion a rather accurate value has been determined for the angular frequency (and by extension, the period). Note that the ratio of the amplitudes of the first two harmonics is small.

9.7 9.7.1

HARMONIC BALANCE: RATIONAL APPROXIMATION Methodology

The standard harmonic balance method is based on constructing truncated approximations to the Fourier series of periodic x(t) and, as a consequence, contains only a finite number of harmonics. An alternative technique is to start with a rational form for x(t), such as xR (t) =

A1 cos θ , 1 + B1 cos 2θ

θ = Ω1 t,

(9.181)

where (A1 , B1 , Ω1 ) are to be determined. The general initial conditions to be satisfied are taken to be xR (0) = A,

x˙ R (0) = 0.

(9.182)

Note that this approximation for x(t) contains all the odd harmonics. In fact, it can be demonstrated that xR (t) =

∞ X

k=0

ak cos(3k + 1)θ,

θ = Ω1 t,

(9.183)

150 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS where "

(−1)k A1 ak = p 1 − B12

#"

1−

#k " # p p B1 − 1 + 1 − B12 1 − B12 . B1 B1

(9.184)

Observe that this expression and that in Equation (9.182) only makes sense if |B1 | < 1.

(9.185)

|B1 | ≪ 1,

(9.186)

If we assume then the coefficients ak decrease exponentially with k, i.e., let “a” be defined as |B1 | e−a = , (9.187) 2 then ( (−1)k A1 e−ak , 0 < B1 ≪ 1, ak = (9.188) A1 e−ak , 0 < (−B1 ) ≪ 1. For purposes of the harmonic balance, x¨R (t) must be calculated. A straightforward, but lengthy calculation gives (    11 3 2 B12 cos θ (1 + B1 cos 2θ) x ¨R = −(Ω1 A1 ) (1 + B1 ) − 2 )   3B1 + 3B1 − 1 cos 3θ + HOH . (9.189) 4 We now illustrate the use of this harmonic balance procedure by applying it to two examples.

9.7.2

x ¨ + x3 = 0

To begin, if we rationalize Equation (9.181) and the cube of the resulting expression, we obtain (1 + B1 cos 2θ)3 x3R = (A1 cos θ)3  3  3 A1 3A1 cos θ + cos 3θ. = 4 4

(9.190)

This result and that from Equation (9.189) allow us to write x¨ + x3 = 0,

(9.191)

APPLICATIONS AND ADVANCED TOPICS

151

as       3A31 11 B12 + cos θ −(Ω21 A1 ) (1 + B1 ) − 2 4     3B1 A3 + −(Ω21 A1 )(3B1 ) − 1 + 1 cos 3θ 4 4 + HOH ≃ 0.

(9.192)

Setting the coefficients of cos θ and cos 3θ to zero gives the two expressions       11 3 2 Ω1 (1 + B1 ) − B12 = A21 , (9.193a) 2 4   3B1 A2 Ω21 (3B1 ) − 1 = 1. (9.193b) 4 4 Dividing these two equations gives a relationship involving only B1 , i.e.,   49 B12 − 10B1 − 1 = 0. (9.194) 4 Solving for B1 gives, for the root having the smallest magnitude, the value B1 = −0.090064. Now, solving Equation (9.193b) for Ω21 , i.e.,
2 3 A1 2 4
2 . Ω1 =  (1 + B1 ) − 11 2 B1

(9.195)

(9.196)

Using the above calculated value of B1 gives for the Ω1 the value Ω1 = (0.930982)A1.

(9.197)

If we take the initial conditions to be x(0) = 1,

x(0) ˙ = y(0) = 0,

(9.198)

then from Equation (9.181) we obtain A1 = 1 + B1 ,

(9.199)

Ω1 = 0.847134,

(9.200)

and with the period T1 =

2π = 7.4170. Ω1

(9.201)

152 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS The exact period for this example, with the initial conditions stated in Equation (9.198), is Texact = 7.4163, (9.202) and, therefore, the percentage error in our calculated period is Texact − T Texact · 100 = 0.01%,

(9.203)

which, for most practical purposes, is very small. To this first-order of approximation, the rational harmonic balance procedure gives, for x ¨ + x3 = 0, the following result xR (t) =

9.7.3

(0.909936) cos[(0.847134)t] . 1 − (0.090064) cos[(1.694268)t]

(9.204)

x ¨ + x2 sgn(x) = 0

The quadratic nonlinear oscillator can be written in two equivalent ways x ¨ + |x|x = 0,

(9.205a)

2

x ¨ + |x| sgn(0) = 0.

(9.205b)

We will construct an approximation to the periodic solution using the rational harmonic balance technique. Writing this approximation as xR (t) =

A1 cos θ , 1 + B1 cos 2θ

θ = Ω1 t,

(9.206)

our task is to determine A1 , B1 , and Ω1 , subject to the initial conditions xR (0) = 1,

x˙ R (0) = yR (0) = 0.

Note that a first integral or Hamiltonian for this system is     1 1 1 2 y + |x|3 = , H(x, y) = 2 3 3 which, on comparing with Equation (9.1) gives p = 2 and q = 3. Assuming 0 < |B1 | < 1, then |xR (t)| =

A1 |cos θ| 1 + B1 cos 2θ

where the |cos θ| has the Fourier expansion          1 1 1 4 + cos 2θ − cos 4θ + · · · . |cos θ| = π 2 3 15

(9.207)

(9.208)

(9.209) (9.210)

(9.211)

APPLICATIONS AND ADVANCED TOPICS

153

Therefore, (  2 8A 3B1 1 3 (1 + B1 cos 2θ) |xR |xR = 1+ cos θ 3π 5 )    17B1 1 1+ cos 3θ + HOH . + 5 7 

(9.212)

Substituting Equations (9.190) and (9.212) into Equation (9.205a) and equating to zero the coefficients of the resulting expressions in cos θ and cos 3θ, gives the following two expressions to be solved for B1 and Ω1 , in terms of A1 :  2      8A1 11 3B1 1+ = Ω21 A1 (1 + B1 ) − B12 , (9.213) 3π 5 2     2  9 17B1 8A1 (9.214) 1+ = Ω21 A1 B12 − 3B1 . 15π 7 4 Dividing these two equations, with simplification of the resulting expression gives       563 169 129 3 2 B1 + B1 − B1 − 1 = 0, (9.215) 28 14 7 for which the smallest-magnitude real solution is B1 = −0.052609.

(9.216)

We also have, from xR (0) = 1, A = 1 + B1 ,

(9.217)

Ω1 = 0.95272,

(9.218)

and can use these to calculate

and

(0.9474) cos[(0.9527)t] . 1 − (0.0526) cos[(1.9054)t] The exact period for this problem is     3 1 2π 1 Γ = Texact = (21/6 ). Ωexact π 3 xR (t) =

9.8

(9.219)

(9.220)

ITERATION METHODS

Nonlinear differential equations modeling periodic systems may also be solved by constructing iteration schemes to obtain approximations to their solutions. We now provide an outline as to how this can be done.

154 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS First, we formulate a direct iteration procedure. The general idea is that the original nonlinear differential equation is transformed into an infinite system of linear inhomogeneous equations such that at a particular step of the calculation, knowledge of the solutions of the previous members of the sequence determines the inhomogeneous term. The basic steps to accomplish these goals are now listed: (a) Assume the differential equation F (¨ x, x) = 0,

x(0) = A,

x(0) ˙ = 0,

(9.221)

is of odd parity and assume that it can be rewritten to a form (which may not be unique) x ¨ + f (¨ x, x) = 0. (9.222) (b) Next add Ω2 x to both sides of Equation (9.222) and obtain x ¨ + Ω2 x = Ω2 x − f (¨ x, x) ≡ G(¨ x, x). Note that, at this point, Ω2 is not known. (c) The iteration scheme is given by ( x ¨k+1 + Ω2k xk+1 = G(¨ xk , xk ), k = 0, 1, 2, . . . , x0 (t) = A cos(Ω0 t),

(9.223)

(9.224) (9.225)

where the xk+1 (t) satisfy the initial conditions xk+1 (0) = A,

x˙ k+1 (0) = 0.

(9.226)

(d) At each step of the iteration, Ωk is determined by the requirement that secular terms do not appear in the solution for xk+1 (t), i.e., xk+1 (t) is periodic and bounded. Note that at step-(k + 1), Ωk appears in Equation (9.224) and, further, at the next step of the iteration (k + 2), Ωk+1 appears, but is unknown and must be calculated. The worked example will help to clarify this point. Comments: (1) The inhomogeneous term in the linear differential equation for xk+1 (t), depends only on the solutions for k less than k + 1, and consequently, are known. (2) The linear differential equation for xk+1 (t) allows the determination of Ωk by requiring that secular terms be absent. Therefore, the angular frequency, “Ω,” appearing in x ¨k and xk in the inhomogeneous term for Equation (9.224) is Ωk . (3) Since, Equation (9.222) is of odd parity, the calculated xk will only contain odd multiples of the angular frequency.

APPLICATIONS AND ADVANCED TOPICS

9.8.1

155

Direct Iteration Scheme: x¨ + x3 = 0

A possible direct iteration scheme for x¨ + x3 = 0,

(9.227)

x ¨k+1 + Ω2k xk+1 = Ω2k xk − x3k .

(9.228)

is Since, x0 (t) = A cos(Ω0 t), where Ω0 , for the time being, is not known, Equation (9.228), for k = 0, is x ¨1 + Ω20 x1 = Ω20 x0 − x30 = Ω20 (A cos θ) − (A cos θ)3      3 3 A 2 2 = Ω0 − A A cos θ − cos 3θ, 4 4

(9.229)

where θ = Ω0 t. Secular terms will not occur in the solution for x1 (t) if the coefficient of the cos θ term is zero, i.e.,   3 A2 = 0, (9.230) Ω20 − 4 which applies Ω0 = Therefore, Equation (9.229) becomes

r

x ¨1 + Ω0 x1 = −

3 A. 4



A3 4



(9.231)

cos 3θ.

(9.232)

The general solution of this last equation is (h)

(p)

x1 = x1 + x1 , where

(h)

( (p)

(9.233)

(h)

x1 = C cos θ,
(p) A x1 = 24 cos 3θ,

(9.234)

and x1 and x1 denote, respectively, the homogeneous and particular solutions, with C an integration constant. Since x1 (0) = A, it follows that   23 A, (9.235) C= 24 and x1 is 

    23 1 x1 (t) = A cos θ + cos 3θ , 24 24 r 3 (At). θ = Ω0 t = 4

(9.236a) (9.236b)

156 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS Thus, the first iteration approximation to the periodic solution of x ¨ + x3 = 0, x(0) = A, and x(0) ˙ = 0 is "  !   !# r r 23 1 3 3 x1 (t) = A cos cos 3 At + At . (9.237) 24 4 24 4 To extend the calculation to k = 1, we use x ¨2 + Ω21 x2 = Ω21 x1 − x31 ,

(9.238)

where x1 (t) takes the form of Equation (9.236a), but now θ is equal to Ω1 t, i.e.,      1 23 cos(Ω1 t) + cos(3Ω1 t) x1 = A 24 24      23 1 =A cos θ + cos(3θ) . (9.239) 24 24 If this expression for x1 is substituted into Equation (9.238), if the resulting coefficient of cos θ is set to zero, and if the complete solution to x2 is then calculated, the following results are obtained, after a long series of algebraic manipulations:  x2 (t) = A (0.955) cos θ + (4.29) · 10−2 cos 3θ + (1.73) · 10−3 cos 5θ + (3.60) · 10−5 cos 7θ + (3.13) · 10−7 cos 9θ , θ = Ω1 (A)t = (0.849326)At.

(9.240a) (9.240b)

Observe that from   Ω0 (A) = (0.866025)A, Ω1 (A) = (0.8493276)A,   Ωexact (A) = (0.847213)A,

we calculate the corresponding percentage errors to be Ωexact (A) − Ω0 (A) · 100 = 2.2%, Ωexact (A) Ωexact (A) − Ω0 (A) = 0.2%. Ωexact (A)

(9.241)

(9.242a) (9.242b)

Our conclusion is that the second-order iteration gives an order of magnitude decrease in the percentage error for the angular frequency.

APPLICATIONS AND ADVANCED TOPICS

9.8.2

157

Extended Iteration: x¨ + x3 = 0

The direct iteration scheme, given by Equation (9.224), can be generalized to the following extended iteration formulation x ¨k+1 + Ω2k xk+1 = G(¨ xk−1 , xk−1 ) + Gx (¨ xk−1 , xk−1 )(xk − xk−1 )

+ Gx¨ (¨ xk−1 , xk−1 )(¨ xk − xk−1 ),

(9.243)

where k = 0, 1, 2, . . . , and Gx ≡

∂G , ∂x

Gx¨ ≡

∂G , ∂x ¨

(9.244)

and the xk+1 satisfy the initial conditions xk+1 (0) = A,

x˙ k+1 (0) = 0.

(9.245)

Inspection of Equation (9.243) indicates that this iteration method requires a knowledge of x−1 (t) and x0 (t), and these are taken to be x−1 (t) = x0 (t) = A cos(Ω0 t).

(9.246)

However, a computationally better scheme to use is the following modification of Equation (9.243). x ¨k+1 + Ω2k xk+1 = G(¨ x0 , x0 ) + Gx (¨ x0 , x0 )(xk − x0 ) + Gx¨ (¨ x0 , x0 )(¨ xk − x ¨).

(9.247)

Therefore, for x ¨ + x3 = 0, the extended iteration scheme, as represented by Equation (9.247), is x ¨k+1 + Ω2k xk+1 = G(x0 ) + Gx (x0 )(xk − x0 )

= (Ω2k x0 − x30 ) + (Ω2k − 3x20 )(xk − x0 ),

(9.248)

where, and this is important, x0 (t) = A cos θ,

θ = Ωk t.

(9.249)

For k = 1, we have x¨2 + Ω21 x2 = (Ω21 x0 − x30 ) + (Ω21 − 3x20 )(x1 − x0 ), where

  x0 (t) = cos θ,
x1 (t) = A 23 24 cos θ +   θ = Ω1 t.

1 24




 cos 3θ ,

(9.250)

(9.251)

158 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS A direct, but lengthy calculation gives Ω1 (A) = (0.846990)A,

(9.252)

(

   13, 244 595 cos θ − cos 3θ 12, 672 12, 672 )   23 + cos 5θ . 12, 672

x2 (t) = A

(9.253)

Two comments need to be made. First, the fractional error for Ω1 (A) in Equation (9.252) is very small, i.e., Ωexact − Ω1 (9.254) Ωexact · 100 = 0.03%. Second, the ratio of the neighboring coefficients is of order 10−2 , i.e.,  | a1 | = 595 ≈ (4.5) · 10−2 , a0 13,244 | a2 | = a1

23 595

≈ (3.9) · 10−2 ,

(9.255)

and these results suggest that the extended iteration scheme may have higher harmonic coefficients, which rapidly decrease to zero. Clearly, the extended scheme is more accurate than its direct version.

9.9

DISCUSSION

It should be noted that for all the applications considered in this chapter, the equation of motions have the form d2 x + a|x|α sgn(x) = 0, dt2

(9.256)

where the parameter a is positive, and α ≥ 0. Written as a system of coupled, first-order, differential equations, we have dx = y, dt

dy = −a|x|α sgn(x). dt

(9.257)

We also used the following initial conditions x(0) = 1,

y(0) =

dx(0) = 0, dt

(9.258)

APPLICATIONS AND ADVANCED TOPICS and this corresponds to the Hamiltonian function   y2 a a H(x, y) = |x|α+1 = + . 2 α+1 α+1

159

(9.259)

An interesting and very important feature of the solutions to Equation (9.256) is that all are periodic, and the exact value of the period is   √ √ 1 4 2π 1 + α Γ 1+α h i . T = (9.260) (1−α) (1 − α)Γ 2(1+α)

Further, this differential equation is of odd parity. This implies that with the initial conditions stated in Equation (9.258), the Fourier series for x(t) has the form ∞ X x(t) = ak cos[(2k + 1)Ωt], (9.261) k=0

where Ω = 2π/T . A comparison of Equations (9.1) and (9.259) shows that p = 2 and q = α + 1. (Note that renormalizing the variables x and y, Equation (9.259) can always be transformed into the expression of Equation (9.1).) The reason why p = 2 is because the systems investigated in this chapter model phenomena appearing in the physical universe and, for such systems, the Newton force law equation is the starting point. Since the Newton force law takes the form ( Force = (mass) acceleration, (9.262) F = ma,

then, for a point particle in one-space dimension, the acceleration can be expressed as d2 x(t) . (9.263) a≡ dt2 Defining the momentum as momentum ≡ m

dx =p dt

(9.264)

and using units of mass such that m = 1, then p(t) =

dx(t) = y(t). dt

(9.265)

Consequently, in the x-y phase-space, dy dy d2 x = =y . dt2 dt dx

(9.266)

160 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS If the system is conservative, then the force depends only on x, i.e., Force = F (x),

(9.267)

dy . dx

(9.268)

and it follows that F (x) = y

Integrating this expression gives ( 2 y constant 2 + U (x) = R U (x) = − F (x)dx.

(9.269)

If U (x) is selected to be

U (x) =

a|x|α+1 , (α + 1)

(9.270)

then with the initial conditions given in Equation (9.258), we arrive at the result of Equation (9.259), and comparing with Equation (9.1), we find p = 2,

q = α + 1.

(9.271)

A detailed examination of the Hamiltonian function, Equation (9.259), shows that it corresponds to a closed curve in the x-y phase-space if α > −1.

(9.272)

Thus, a system modeled by the differential equation d2 x 1 + 1/3 = 0, 2 dt x

(9.273)

has bounded phase-space curves and all solutions are periodic; observe that the Hamiltonian function (
2 H(x, y) = y2 + 32 x2/3 = 32 , (9.274) x(0) = 1, y(0) = 0. However, α = −1 requires a very detailed and careful analysis to prevent difficulties from arising. Another way of characterizing Equation (9.256) is to observe that it is a “truly nonlinear (TNL)” oscillator differential equation; see Section 9.3. Equations of this type have the feature that no “small parameter” can be introduced such that any of the standard perturbation techniques may be applied to calculate analytical approximations to their periodic solutions. There are basically two ways to get around this limitation. First, we can numerically integrate the equation of motion, Equation (9.257). Second, schemes can be created which allow the determination of analytical approximations. Two

APPLICATIONS AND ADVANCED TOPICS

161

particular useful procedures are the method of harmonic balance and iteration techniques. Sections 9.6 and 9.7, and 9.8 respectively, deal with these two methodologies. With regard to harmonic balance methods, generally, an increase in the number of Fourier harmonic terms increases the precision of the calculated x(t) and the values of their periods. However, iteration methods face some potential difficulties. The main problem being the selection of a scheme to be iterated. This is currently an active area of research. Finally, there are the so-called Ateb functions corresponding to the exact solutions of Equation (9.256). While there has been some work done on constructing analytical approximations to the Ateb functions, it is not entirely clear (to me) that this provides generally a better approach than the harmonic balance methods. Additional investigations are required to give a definitive resolution of these issues.

NOTES AND REFERENCES Section 9.1: Excellent introductions to nonlinear oscillators are given in the following three books: (1) R. E. Mickens, Introduction to Nonlinear Oscillations (Cambridge University Press, New York, 1981). (2) I. Kovacic and M. J. Brennan, The Duffing Equation: Nonlinear Oscillators and their Behaviors (Wiley, New York, 2011). (3) L. Cveticanin, Strong Nonlinear Oscillators: Analytical Solutions (Springer, Berlin, 2017). Section 9.2: This section is based on the paper (1) R. E. Mickens, Fourier representations for periodic solutions of oddparity systems, Journal of Sound and Vibration 258 (2002), 398–401. Section 9.3: A full discussion of the concept of “truly nonlinear (TNL)” oscillators is the book (1) R. E. Mickens, Truly Nonlinear Oscillations: Harmonic Balance, Parameter Expansions, Iteration, and Averaging Methods (World Scientific, Singapore, 2010). Section 9.4: The classical paper on Ateb periodic functions is (1) R. M. Rosenberg, The Ateb(h)-functions and their properties, Quarterly of Applied Mathematics 21 (1963), 37–47.

162 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS See also the book by Cveticanin (Section 9.1, above), Appendices A and B, and the following papers: (2) I. V. Andrianov, V. I. Olevski, and Y. B. Olveska, Analytical approximation of periodic Ateb-functions via elementary functions, Proceedings of the 5th International Conference on Nonlinear Dynamics (NDKhPI2016, September 27–30, 2016, Kharkov, Ukraine), 260–267. (3) I. Kovacic, On the response of purely nonlinear oscillators: An Ateb-type solution for motion and an Ateb-type external excitation, International Journal of Non-Linear Mechanics 92 (2017), 15–24. Section 9.5: This section is based on my work as presented in the paper (1) R. E. Mickens, A note on exact finite difference schemes for the differential equations satisfied by the Jacobi cosine and sine functions, Journal of Difference Equations and Applications 19 (2013), 1042–1047. See also the following: (2) R. B. Potts, Best difference equation approximation to Duffing’s equation, Journal of Austria Mathematics Society 23B (1982), 349–356. (3) R. E. Mickens, Nonstandard finite-difference schemes for differential equations, Journal of Difference Equations and Applications 8 (2002), 823–847. Sections 9.6 and 9.7: The discussions in these sections are based closely on the following papers: (1) R. E. Mickens, Comments on the method of harmonic balance, Journal of Sound and Vibration 94 (1984), 456–460. (2) R. E. Mickens, A generalization of the method of harmonic balance, Journal of Sound and Vibration 111 (1986), 515–519. (3) R. E. Mickens and D. Semwogerere, Fourier analysis of a rational harmonic balance approximation for periodic solutions, Journal of Sound and Vibration 195 (1996), 528–530. Section 9.8: Good discussions of the general methodology of iteration techniques applied to nonlinear oscillators can be found in the following articles: (1) R. E. Mickens, Iteration procedure for determining approximate solutions to nonlinear oscillator equations, Journal of Sound and Vibration 116 (1987), 185–190. (2) R. E. Mickens, A generalized iteration procedure for calculating approximations to periodic solutions of “truly nonlinear oscillators,” Journal of Sound and Vibration 287 (2005), 1045–1051.

APPLICATIONS AND ADVANCED TOPICS

163

Section 9.9: A general analysis of “truly nonlinear (TNL)” oscillators and various techniques to obtain approximations to their solutions is (1) R. E. Mickens, Truly Nonlinear Oscillations: Harmonic Balance, Parameter Expansions, Iteration, and Averaging Methods (World Scientific, London, 2010). A rather large set of works has been devoted to the case of fractional power nonlinearities. Details and summaries are given in the following two articles: (2) R. E. Mickens, Oscillations in an x4/3 potential, Journal of Sound and Vibration 246 (2001), 375–378. (3) H. Gottlieb, Frequencies of oscillators with fractional-power nonlinearities, Journal of Sound and Vibration 261 (2003), 557–566. Examples of such oscillators are x¨ +

1 x1/3

= 0 or x¨ + x1/3 = 0.

The oscillator

1 = 0, x has been investigated by Mickens and several other researchers; see, for example, the following: x ¨+

(4) R. E. Mickens, Harmonic balance and iteration calculations of periodic solutions to y¨ + y −1 = 0, Journal of Sound and Vibration 306 (2007), 398–401. (5) Johanna Denise Garcia Saldaña, A Qualitative and Quantitative Study of Some Planar Differential Equations – A thesis submitted in fulfillment of the requirements for the degree of Doctor of Mathematics, Departament de Matemàtiques, Universitat Autònoma de Barcelona (February 2014). Ateb-type periodic functions are being used to help in the representation of the periodic solutions of strongly nonlinear oscillators. Since the Ateb functions cannot be expressed in terms of a finite combination of elementary functions, approximations are required. An attempt in this direction is the paper (6) I. V. Andrianov, V. I. Olevskyi, and Y. B. Olsevska, Analytical approximation of periodic Ateb-functions via elementary functions, Proceedings of the 5th International Conference on Nonlinear Dynamics (NDKhPI2016, September 27–30, 2016, Kharkov, Ukraine), pp. 260–267.

Chapter 10 FINALE

10.1

GOALS

The major goals of this book were to present and discuss some preliminary investigations on new families of periodic functions and, in some cases, also examine the properties of the associated hyperbolic functions. This work centered on an examination of the formulas |y|p + |x|q = 1,

x(0) = 1,

y(0) = 0,

(10.1)

|y|p − |x|q = 1,

x(0) = 0,

y(0) = 1,

(10.2)

where, in general, p > 0,

q > 0.

(10.3)

From our perspective, what we have given is an extension and new interpretation of previous results obtained on the so-called “generalized trigonometric functions.” The essential features of our work derive from considering the expressions in Equations (10.1) and (10.2) from three different aspects: A: as geometrical structures in either the (x − y) or (r − θ) planes, B: as a Hamiltonian function for a one-space-dimensional dynamical system and C: as functional equations.

10.2

RESULTS

The following is a summary of the major results obtained from the various methodologies used in this book: 1) Equation (10.1) has periodic solutions. For the geometrical case, where the independent variable is θ, 0 ≤ θ ≤ 2π, these solutions are periodic, with period 2π. Further, three classes of periodic solutions arise and these functions 165

166 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS are direct generalizations of the standard trigonometric cosine, sine, and dine functions. 2) If the expression in Equation (10.1) is taken to be the Hamiltonian of a dynamical system, for which t, the time, is the independent variable, then x(t) and y(t) are periodic functions of t. 3) If now Equation (10.1) is considered to be a functional equation, then it has many families of solutions, some of which are periodic. 4) For Equation (10.2), there are two major families of solutions. First, with θ, 0 ≤ θ ≤ 2π, as the independent variable, periodic solutions exist, but are not defined over specific intervals of x. Also these periodic functions may become unbounded for particular values of θ. 5) Considered as a dynamical system, Equation (10.2) can have continuous solutions which become unbounded as t → (±)∞. 6) Many of the most important applications in the natural and engineering sciences correspond to p = 2, q > 0, (10.4) and this result is a consequence of the Newton definition of force. 7) The expressions in Equations (10.1) and (10.2) are invariant under the transformations   T1 : x → −x, y → y, (10.5) T2 : x → x, y → −y,   T3 = T1 T2 = T2 T1 : x → −x, y → −y,

and, as a consequence, both the periodic and non-periodic solutions have many common features imposed on their respective solutions, either x(θ) and y(θ), or x(t) and y(t).

10.3

SOME UNRESOLVED TOPICS AND ISSUES

We end the book with a brief presentation of various currently unresolved topics and related issues, all of which are of importance in research efforts directed toward the creation, analysis, and understanding of generalized trigonometric functions. (I) Equations (10.1) and (10.2) were originally formulated for real, nonnegative values of p and q. One consequence of doing this is that absolutevalues must appear. However, in actual physical universe-related matters, where these equations may appear, they can only be known to a finite number of numerical places. For such cases, it makes (experimental) sense to have

FINALE

167

rational representations for p and q. One possibility is to use  2n1  p ≡ 2m1 +1 , n1 = 1, 2, 3, . . . ,   m1 = 0, 1, 2, . . . ,

and

Note that

and

(

 2n2  q ≡ 2m2 +1 , n2 = 1, 2, 3, . . . ,   m2 = 0, 1, 2, . . . .

p−1= q−1=

2n1 2m1 +1 2n2 2m2 +1

−1=

−1=

2(n1 −m1 )−1 , 2m1 +1 2(n2 −m2 )−1 , 2m2 +1

    n1 > m1 + 12 (p − 1) > 0 ⇒ . (q − 1) > 0 n2 > m2 + 21

(10.6)

(10.7)

(10.8)

(10.9)

(II) A major extension of the geometrical approach would be to consider general simple closed, convex curves, in the x-y plane, for which the origin lies within the curves. We expect the existence of three types of periodic functions similar to the trigonometric cosine, sine, and dine functions. However, for an arbitrary curve, the details of the analysis would be expected to be much more complex than what we have done in our work. (III) The geometrical periodic functions associated with x2 + y 2 = 1, and

(10.10)

 x 2

+ y 2 = 1, a > 1, (10.11) a differ greatly in the functions they generate. For Equation (10.10), the associated differential equation is d2 x + u = 0, du2

(10.12)

while for the second equation, we have d2 x + Ax + Bx3 = 0, du2

(10.13)

where A and B are constants, related to a. Inspection of these differential equations shows one to be linear, the second to be nonlinear. Consider a simple, closed, convex curve, surrounding the origin, taking the form f (x, y, a) = 0, (10.14)

168 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS where a is a parameter. The results from Equations (10.12) and(10.13) indicate that at specific values of the parameter a, a “bifurcation” in the structure of an associated differential equation may take place. Is this a general phenomenon? (IV) Do “interesting” and useful solutions for the modeling of dynamical systems exist for Equation (10.2) which are not (asymptotically) power-law or exponential in mathematical structure? (V) A general issue: Is there a finite set of families of periodic functions, each with a possible infinite number of members? (VI) For the case p = 2, q > 1, it would be of value for the techniques such as the method of harmonic balance to determine the exact Fourier series of the following expressions  cos θ+A1 cos 3θ   1+B1 cos 2θ+B2 cos 4θ , cos θ (10.15) , 1 cos 2θ+C2 sin 2θ   1+C α |x| sgn(x), where (A1 , B1 , B2 , C1 , C2 ) are parameters such that B1 + B2 < 1,

C1 + C2 < 1,

(10.16)

and α > 0.

(10.17)

(VII) For Equation (10.1), with p = 2, can new methodologies be formulated for the construction of approximations to the exact solutions of the equations of motion? (This means that Equation (10.1) is to be taken as a Hamiltonian function.) This issue involves the creation of extensions to the standard procedures for harmonic balance and iteration schemes. (VIII) Can generalized Ateb-type periodic functions be constructed for  p q  |y| + |x| = 1, x(0) = 1, y(0) = 0,   p ≥ 1, q ≥ 1?

NOTES AND REFERENCES Section 10.3: Issue (I) is introduced and briefly discussed in the following book: (1) R. E. Mickens, Truly Nonlinear Oscillations (World Scientific, London, 2010). Note that pe ≡

2n , 2m + 1

po ≡

2n + 1 , 2m + 1

FINALE

169

where (m, n) are non-negative integers having the properties (−x)pe = xpe ,

(−x)po = −(x)po .

Issue (III) may be related to an essential feature in the formulation of the equations of motion for the Hamiltonian formulation, i.e., for example, if   1 1 H(x, y) = (x2 + y 2 ) = , 2 2 then the equations of motion are ∂H dx = = y, dt ∂y

dy ∂H =− = −x. dt ∂x

However, dx dy dH(x, y) =x +y = 0, dt dt dt and the following three possibilities may occur: dy dx = y, = −x, dt dt dx dy B: = , x = −y, dt dt dy dx = φ(x, y, t)y, = −φ(x, y, t)x, C: dt dt

A:

where φ(x, y, t) is an arbitrary function of its arguments. Case A is the standard answer for the equations of motion. Case B implies x(t) = y(t) = 0, and consequently, is not of interest. However, Case C is a “nonstandard” representation and reduces to Case A when φ(x, y, t) = 1. Issue (VII) appears in the book by Mickens, see (1) above. Issue (VIII) seeks to extend the various investigations of the equation y 2 + |x|q = 1,

q ≥ 1,

to the general case |y|p + |x|q = 1,

p ≥ 1,

q ≥ 1.

See the following references: (2) R. M. Rosenberg, The Ateb(h) functions and their properties, Quarterly Journal of Applied Mathematics 21 (1963), 37–47. (3) A. Beléndez, J. Francés, T. Beléndez, S. Bleda, C. Pascual, and E. Arribas, Nonlinear oscillator with power-form elastic form elastic-term: Fourier series expansion of the exact solution, Communications in Nonlinear Science and Numerical Simulation 22 (2015), 134–148.

170 GENERALIZED TRIGONOMETRIC & HYPERBOLIC FUNCTIONS (4) I. Kovacic, On the response of purely nonlinear oscillators: An Ateb-type solution for motion and an Ateb-type external excitation, International Journal of Non-Linear Mechanics 92 (2017), 15–24.

APPENDIX

A TRIGONOMETRIC RELATIONS A.1 Basic Properties (sin θ)2 + (cos θ)2 = 1 −1 ≤ sin θ ≤ +1,

−1 ≤ cos θ ≤ +1

sin(−θ) = − sin θ,

cos(−θ) = cos θ

sin(θ + 2π) = sin θ,

cos(θ + 2π) = cos θ

  sin(0) = 0,  
 π   sin 2 = 1, sin(π) = 0, 
  = −1, sin 3π  2   sin(2π) = 1,

cos(0) = 1
cos π2 = 0 cos(π) = −1,
cos 3π =0 2 cos(2π) = 1

A.2 Exponential Definitions of Trigonometric Functions sin θ =

eiθ − e−iθ , 2i

cos θ =

i=

√ −1,

eiθ + e−iθ , 2

and eiθ = cos θ + i sin θ e−iθ = (eiθ )∗ = cos θ − i sin θ. A.3 Functions of Sums of Angles sin(θ1 ± θ2 ) = sin θ1 cos θ2 ± cos θ1 sin θ2 cos(θ1 ± θ2 ) = cos θ1 cos θ2 ± sin θ1 sin θ2

171

172

APPENDIX

A.4 Powers of Trigonometric Functions   1 2 sin θ = (1 − cos 2θ) 2   1 (1 + cos 2θ) cos2 θ = 2   1 sin3 θ = (3 sin θ − sin 3θ) 4   1 cos3 θ = (3 cos θ + cos 3θ) 4   1 (3 − 4 cos 2θ + cos 4θ) sin4 θ = 8   1 cos4 θ = (3 + 4 cos 2θ + cos 4θ) 8   1 (10 sin θ − 5 sin 3θ + sin 5θ) sin5 θ = 16   1 cos5 θ = (10 cos θ + 5 cos 3θ + cos 5θ) 16 A.5 Other Trigonometric Relations    θ1 ∓ θ2 θ1 ± θ2 cos 2 2     θ1 − θ2 θ1 + θ2 cos = 2 cos 2 2     θ1 + θ2 θ1 − θ2 = −2 sin sin 2 2   1 = [sin(θ1 + θ2 ) + sin(θ1 − θ2 )] 2   1 [sin(θ1 + θ2 ) − sin(θ1 − θ2 )] = 2   1 = [cos(θ1 + θ2 ) + cos(θ1 − θ2 )] 2   1 [cos(θ1 + θ2 ) − cos(θ1 − θ2 )] = 2

sin θ1 ± sin θ2 = 2 sin cos θ1 + cos θ2 cos θ1 − cos θ2 sin θ1 cos θ2 cos θ1 sin θ2 cos θ1 cos θ2 sin θ1 sin θ2



APPENDIX

173

A.6 Derivatives and Integrals d cos θ = − sin θ, dθ d2 cos θ = − cos θ, dθ2

d sin θ = cos θ dθ d2 sin θ = − sin θ dθ2

Both cos θ and sin θ are solutions to the following second-order, linear differential equation d2 u(θ) + u(θ) = 0, dθ2 du(0) u(0) = 1, = 0, gives u(θ) = cos θ, dθ du(0) = 1, gives u(θ) = sin θ. u(0) = 0, dθ Also Z cos θdθ = sin θ + C1 , Z

sin θdθ = (−) cos θ + C2 ,

where C1 and C2 are arbitrary constants. A.7 Taylor Series cos θ =

∞ X

(−)n

n=0

θ2 θ4 θ2n + + · · · + (−)n + ··· 2! 4! (2n)!

=1− sin θ =

∞ X

(−)n

n=0

=θ−

θ2n (2n)!

θ2n+1 (2n + 1)!

θ5 θ2n+1 θ3 + + · · · + (−)n + ··· 3! 5! (2n + 1)!

A.8 Other Derived Trigonometric Functions tangent : cotangent : secant : cosecant :

tan(θ) cotan(θ) sec(θ) cosec(θ)

sin θ cos θ cos θ ≡ sin θ 1 ≡ cos θ 1 ≡ sin θ ≡

174

APPENDIX

B TRIGONOMETRIC HYPERBOLIC RELATIONS B.1 Exponential Definition of Hyperbolic Functions ex − e−x 2 ex + e−x Hyperbolic cosine : cosh(x) ≡ 2 Hyperbolic sine : sinh(x) ≡

Therefore, it follows that ex = cosh(x) + sinh(x) e−x = cosh(x) − sinh(x). The sinh(x) and cosh(x) satisfy the relation cosh2 (x) − sinh2 (x) = 1. B.2 Basic Properties cosh(−x) = cosh(x), cosh(0) = 1,

sinh(−x) = − sinh(x) sinh(0) = 0

Lim cosh(x) = +∞

x→±∞

Lim sinh(x) = (±)∞

x→±∞

B.3 Derivatives and Integrals d cosh(x) = sinh(x), dx

d sinh(x) = cosh(x) dx

d2 d2 cosh(x) = cosh(x), sinh(x) = sinh(x) dx2 dx2 From these two latter relations, it follows that cosh(x) and sinh(x) are solutions to the second-order, linear differential equation d2 w(x) − w(x) = 0, dx2 such that w(0) = 1, w(0) = 0, Also

Z

dw(0) = 0, dx dw(0) = 1, dx

gives w(x) = cosh(x), gives w(x) = sinh(x).

sinh(x)dx = cosh(x) + C1 ,

APPENDIX Z

175

cosh(x)dx = sinh(x) + C2 ,

where C1 and C2 are arbitrary constants. B.4 Related Hyperbolic Functions sinh(x) cosh(x) cosh(x) Hyperbolic cotangent : coth(x) ≡ sinh(x) 1 Hyperbolic secant : sech(x) ≡ cosh(x) 1 Hyperbolic cosecant : csch(x) ≡ sinh(x) Hyperbolic tangent : tanh(x) ≡

B.5 Sum and Subtraction Formulas sinh(x + y) = sinh(x) cosh(y) + cosh(x) sinh(y) cosh(x + y) = cosh(x) cosh(y) + sinh(x) sinh(y) sinh(x − y) = sinh(x) cosh(y) − cosh(x) sinh(y) cosh(x − y) = cosh(x) cosh(y) − sinh(x) sinh(y)

   x−y x+y cosh 2 2     x−y x+y cosh cosh(x) + cosh(y) = 2 cosh 2 2     x+y x−y sinh(x) − sinh(y) = 2 cosh sinh 2 2     x+y x−y cosh(x) − cosh(y) = 2 sinh sinh 2 2 sinh(x) + sinh(y) = 2 sinh



B.6 Relations between Hyperbolic and Trigonometric Functions Using the exponential functions definitions of the cosine and sine functions, then the following relations exist between the trigonometric and hyperbolic functions: √ sinh(x) = −i sin(ix), i = −1 cosh(x) = cos(ix) tanh(x) = −i tan(ix) coth(x) = i cot(ix) sech(x) = sec(ix) csch(x) = i csc(ix)

176

APPENDIX

B.7 Taylor Series sinh(x) =

∞ X x2n+1 (2n + 1)! n=0

=x+ cosh(x) =

x3 x5 x2n+1 + + ··· + + ··· 3! 5! (2n + 1)!

∞ X x2n (2n)! n=0

=1+

x4 x2n x2 + + ··· + + ··· 2! 4! (2n)!

C SPECIAL FUNCTIONS C.1 Properties of Even and Odd Functions Consider functions f (x) and g(h), defined on a symmetric interval (−a, a), a > 0, where a may be unbounded. • A function f (x) is an even function on the interval (−a, a), if and only if f (−x) = f (x). if

• A function f (x) is an odd function on the interval (−a, a), if and only f (−x) = −f (x).

• Given an arbitrary function g(x) defined on the interval (−a, a), then it can be written as g(x) = g (+) (x) + g (−) (x), where

g(x) + g(−x) , 2 g(x) − g(−x) , g (−) (x) = 2 g (+) (x) =

and g (+) (−x) = g (+) (x),

g (−) (−x) = −g (−) (x).

The functions g (+) (x) and g (−) (x) are the even and odd parts, respectively, of the function g(x). The following propositions are true: a) If f (x) and g(x) are both even functions, then h(x) = f (x)g(x) is an even function. b) If f (x) and g(x) are both odd functions, then h(x) = f (x)g(x) is an even function. c) If f (x) is an even function and g(x) is an odd function, then h(x) = f (x)g(x) is an odd function.

APPENDIX

177

d) Let f (x) be an even function, integrable on the interval, (−a, a); then Z a Z a f (x)dx = 2 f (x)dx. −a

0

e) If f (x) is an odd function, integrable over the interval (−a, a), then Z a f (x)dx = 0. −a

f) Let f (x) be an even (odd) function defined over the interval (−a, a), then its derivative is an odd (even) function on the same interval, i.e., df (x) odd, dx df (x) f (x) odd ⇒ even. dx

f (x) even ⇒

C.2 Absolute Value Function Let a be a nonzero real number. The absolute value of a is defined to be   if a > 0, a, |a| = 0, if a = 0,   −a, if a < 0. An alternative definition is

|a| =

√

a2 .

The following rules apply: i) |a| ≥ 0 ii) | − a| = |a| iii) |ab| = |a||b| iv) ab = |a| |b| , b 6= 0 v) |an | = |a|n vi) |a + b| ≤ |a| + |b|, triangle inequality. The above rules generalize to the situation where a and b are functions. A graph of |x|, the absolute value function, is given in Fig. C.2.1.

178

APPENDIX

x Fig. C.2.1

Plot of the absolute value function, |x|.

C.3 Sign and Theta Functions The theta and sign functions are defined, respectively, as follows ( 1, if x > 0, θ(x) ≡ 0, if x < 0, sign(x) ≡

(

1, if x > 0, −1, if x < 0.

Since both θ(x) and sign(x) are discontinuous at x = 0, they are not defined at this point. However, depending on the circumstances, the following values may be selected 1 θ(0) = , sign(0) = 0. 2 The following relation holds for the theta and sign functions sign(x) = θ(x) − θ(−x). Plots of these functions appear in Fig. C.3.1. Also, note that sign(x) =

d|x| . dx

APPENDIX

179

x

(a) O(x)

1/2

x

(b) Fig. C.3.1 (a) Plot of sign(x) vs x. (b) Plot of θ(x) vs x.

180

APPENDIX

C.4 Delta Function A function, δ(x), is a “delta function” if it formally satisfies the three conditions: R∞ a) −∞ δ(x)dx = 0, b) Rδ(x) = 0, x 6= 0, ∞ c) −∞ δ(x)f (x)dx = f (0). The use of a “delta function” is based on several important properties: d) The “delta function” is even, i.e., δ(−x) = δ(x). e)

R∞

R −∞ ∞

δ(ax)f (x) =



1 |a|



f (0).

f) −∞ δ(x − a)f (x)dx = f (a). This result follows directly from c). g) Let g(x) be a function having a finite number, N , of simple zeros at (x1 , x2 , . . . , xN ). We then have δ[g(x)] =

N X δ(x − xk ) . |g ′ (xk )| k=1

h) Rxδ ′ (x) = −δ(x). ∞ i) −∞ δ (n) (x)f (x)dx = (−)n f (n) (0), where δ (n) (x) ≡

dn δ(x) , dxn

f (n) (x) ≡

dn f (x) . dxn

D FOURIER SERIES Let f (x) be defined on −∞ < x < ∞ and have period 2L, i.e., f (x + 2L) = f (x). If the following integrals exist,   Z 2L kπx dx, f (x) cos L 0

Z

0

2L

f (x) sin



kπx L



dx,

for k = 0, 1, 2, . . . , then the formal Fourier series of f (x) on 0 < x < 2L is given by the expression     ∞  kπx kπx a0 X + bk sin , ak cos + f (x) ∼ 2 L L k=1

where   Z 2L   1 kπx ak = dx, f (x) cos L L 0     Z 2L kπx 1 dx. f (x) sin bk = L L 0

APPENDIX

181

Definition 1: A function f (x) is said to be piecewise continuous on a finite interval, a ≤ x ≤ b, if this interval can be partitioned into a finite number of subintervals such that f (x) is continuous in the interior of each of the subintervals and f (x) has finite limits as x approaches either end point of each subinterval from its interior. Definition 2: A function f (x) is said to be piecewise smooth on a finite interval, a ≤ x ≤ b, if both f (x) and f ′ (x) are piecewise continuous on a ≤ x ≤ b. Theorem 1: Let f (x) be piecewise smooth on the interval, 0 < x < 2L, then its Fourier series is given by     ∞  kπx kπx a0 X + bk sin . + ak cos 2 L L k=1

The Fourier series converge at every point x, in the interval 0 < x < 2L to the value f (x+ ) + f (x− ) , 2 where f (x+ ) is the right-hand limit of the function f at x and f (x− ) is the left-hand limit of the function f at x. If f is continuous at x, then the Fourier series of f at x converges to f (x). E HAMILTONIAN DYNAMICS Consider a dynamic system having N -degrees-of-freedom (NDOF). Assume that there exists a function H, called the Hamiltonian, which depends on 2N + 1 variables, i.e., H(x, p, t) = constant, where

 x1 (t)  x2 (t)    x(t) =  .  ,  ..  

xN (t)

 p1 (t)  p2 (t)    p(t) =  .  .  ..  

(E.1)

(E.2)

pN (t)

The x(t) and p(t) will be called the generalized coordinates and momenta, and they both depend on time, t. It is from the Hamiltonian that all the laws governing the dynamics of the system must have their genesis. In other words, dx(t)/dt and dp(t)/dt must be derivable from a knowledge of H(x, p, t) and some other physical and mathematical requirements. Experience with the physical universe leads to the following conclusions: In general, the outcome of an experiment does not depend on where it is done (location) or when it is done (time). Since the Hamiltonian is assumed to determine the dynamics, it follows from

182

APPENDIX

the second observation that H(x, p, t) must be invariant under an arbitrary time translation, t0 , i.e., t → t + t0 . (E.3) It is sufficient to consider an infinitesimal time translation, t → t + δt. Thus, the time translation invariance can be expressed as the condition

or H(x, p, t) +

H(x, p, t + δt) = H(x, p, t),

(E.4)

∂H(x, p, t) (δt) + O[(δt)2 ] = H(x, p, t), ∂t

(E.5)

and this implies that ∂H(x, p, t) = 0. ∂t The conclusion is that the Hamiltonian does not depend on time H = H(x, p) = constant.

(E.6)

(E.7)

The first observation of space translation invariance, i.e., x → x + δx,

(E.8)

H(x + δx, p) = H(x, p),

(E.9)

gives the condition where δx is the same constant for each coordinate, i.e., xi → xi + a,

(i = 1, 2, . . . N ).

(E.10)

Comment: One has to be careful here. Certainly, if the x’s are components of Cartesian coordinates, then it can always be arranged to have all of the separate (x, y, z) “components” be translated by the same magnitude. For infinitesimal δx = a, we have H(x, p) + a

n X ∂H(x, p) i=1

and it follows that

+ O(a2 ) = H(x, p),

n X ∂H(x, p) i=1

Returning to

∂xi

∂xi

= 0.

H(x, p) = constant,

(E.11)

(E.12)

(E.13)

and taking the total time derivative of H(x, p), gives  n  X dH ∂H dxi ∂H dpi = 0. = + dt ∂xi dt ∂pi dt i=1

(E.14)

APPENDIX

183

This latter equation allows us to determine a connection between the Hamiltonian function, H(x, p), and the evolution equations for the coordinates and the momentum. The direct and simplest possibility is to make the choice ( dxi (t) = ∂H(x,p) (i = 1, 2, . . . , N ), dt ∂pi (E.15) ∂H(x,p) dpi (t) = − ∂xi (i = 1, 2, . . . , N ). dt Therefore, given H(x, p), we have a calculational procedure to determine the equations of motion. They are 2N , coupled, ordinary differential equations for the N -coordinates and N -momenta. Equations (E.15) are the Hamiltonian equations of motion. Finally, comparing Equations (E.12) and (E.15), we conclude that the result in Equation (E.12) is just the conservation of momentum, i.e., N X ∂H(x, p) i=1

∂xi

N X

dpi (t) (−) = =− dt i=1



d dt

X N

pi (t) = 0,

(E.16)

i=1

and this implies N X

pi (t) = constant.

i=1

NOTES AND REFERENCES Sections A and B : Knowledge of the trigonometric periodic and hyperbolic functions is ubiquitous and any of a number of the standard textbooks and handbooks will provide the full details of the properties of these functions and the associated derivations. Section C : The delta function was “created” by P. A. M. Dirac and discussed (non-mathematically) in this book: (i) P. A. M. Dirac, The Principals of Quantum Mechanics, 2nd ed. (The Clarendon Press, Oxford 1930). A formal examination of such “generalized functions” was done by M. J. Lighthill and appears in his book: (ii) M. J. Lighthill, Introduction to Fourier Series and Generalized Functions (Cambridge University Press, London, 1958). For a general, heuristic discussion, see: (iii) R. E. Mickens, Mathematical Methods for the Natural and Engineering Sciences, 2nd ed. (World Scientific, London, 2017). Section D : Two basic, concise, and good introductions to Fourier series are the books: (iv) S. L. Ross, Differential Equations (Blaisdell, Waltham, MA; 1964).

184

APPENDIX

(v) J. D. Logan, Applied Mathematics, A Contemporary Approach, (WileyInterscience, New York, 1987). Section E : One of the best introductions to the Hamiltonian approach to classical mechanics is the book: (vi) H. Goldstein, Classical Mechanics (Addison-Wesley; Cambridge, MA; 1951).

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M. R. Roussel, Hamiltonian Systems: http://people.uleth.ca/roussel/nld/ Hamiltonian.pdf, 2005. W. Schwalm, Elliptic functions sn, cn, dn as trigonometry: http://www.und.edu/instruct/schwalm/MAA_Presentation_10_02/handout.pdf, 2005.

Index

A Absolute value function, 177, 178 Addition relations, 7 Antisymmetric, constant force oscillator, 134–135 Applications and advanced topics, 127–163 antisymmetric, constant force oscillator, 134–135 Ateb periodic functions, 129, 138–140 cube-root oscillator, 147 direct iteration scheme, 155–156 discussion, 158–161 Duffing oscillator, 128 elementary functions, 130 equations of motion, 127 exact difference equation for cn, 141–142 exact difference equation for sn, 142–143 exact discretization of Jacobi elliptic differential equations, 140–143 extended iteration, 157–158 first-order harmonic balance, 145 Hamiltonian, conservative dynamical systems, 127 harmonic balance (direct method), 143–149 harmonic balance (rational approximation), 149–153 harmonic oscillator, 128 inhomogeneous term, 154

iteration methods, 153–158 iteration procedures, 130 Jacobi cosine, 140 Jacobi elliptic function, 137 Newton’s force law, 128 observations, 136 odd-parity systems and their Fourier representations, 130–133 particle in a box, 136 rational procedure, 152 rescaled Duffing equation, 140 restricted Duffing equation, 137 simple harmonic oscillator equation, 140 truly nonlinear oscillators, 129, 133–137, 160 uniqueness requirement, 132 x¨ + x1/3 = 0, 147–149 x¨ + x2 sgn(x) = 0, 152–153 x¨ + x3 = 0, 145–147 x¨ + x3 = 0, 150–152 Ateb periodic functions, 129, 138–140, 168 C Cosine functions, see Sine and cosine functions, trigonometric and hyperbolic Cube-root oscillator, 147 D Direct iteration scheme, 155–156 Duffing equation, 140 Duffing oscillator, 128 189

190 E Elliptic functions, 23–46 additional properties of Jacobi elliptic functions, 34–37 bounds and symmetries, 39 calculation of u(θ) and the period for cn, sn, dn, 32–33 cn, sn, dn for special k values, 36 definition of dynamic system, 37–38 differential equations, 31–32 dynamical system interpretation of elliptic Jacobi functions, 37–40 elliptic Hamiltonian dynamics, 26–28 first derivatives, 30–31 first integrals, 38–39 Fourier series, 36–37 fundamental relations for square of functions, 34 hyperbolic elliptic functions as a dynamic system, 40–41 hyperbolic θ-periodic elliptic functions, 41–45 Jacobi, cn, sn, and dn functions, 28–34 Jacobi elliptic functions, elementary properties of, 29 Jacobi elliptic functions, special values of, 33–34 limits k → 0+ and k → 0− , 38 θ-periodic elliptic functions, 23–26 product relations, 36 second-order differential equations, 40 Even function, 59, 176 Extended iteration, 157–158 F Fourier representation, odd-parity systems and, 130–133 Fourier series, 180–181

Index Jacobi elliptic functions, 36–37 square trigonometric functions, 53–55 G Generalized periodic solutions of f (t)2 + g(t)2 = 1, 83–97 differential equation for f(t) and g(t), 89–92 dine function, 84, 94 discussion, 92–94 example, 88–89 Fourier representation, 87 generalized cosine and sine functions, 84–86 Jacobi elliptic functions, 83, 86, 93 mathematical structure of θ(t), 86–88 non-periodic solutions, 94–97 Generalized trigonometric functions, previous results on, 99–108 area of enclosed curve, 105 beta function, 105 definition as integral forms, 101–102 differential equation formulation, 100–101 gamma function, 105 geometric approach, 102 Hamiltonian formulation, 104 period, 106 Pythagorean identity, 100 sine function, 101 symmetry considerations and consequences, 103–106 symmetry transformation and consequences, 103 Generalized trigonometric functions, |y|p + |x|q = 1, 109–117 gallery of particular solutions, 111–117 methodology, 109–111

Index periodic solutions, 110 transformation of dependent variables, 110 Generalized trigonometric hyperbolic functions, |y|p − |x|q = 1, 119–126 asymptotic relations, 119 gallery of special solutions, 120–126 solutions, 120 Goals, 165 H Hamiltonian classes of motion, 59 conservative dynamical systems, 127 generalized trigonometric functions, 104 harmonic balance (rational approximation), 152 parabolic trigonometric functions, 67 square functions, 56 Hamiltonian dynamics, 181–183 components of Cartesian coordinates, 182 elliptic, 26–28 equations of motion, 183 generalized coordinates, 181 N-degrees-of-freedom, 181 Harmonic balance (direct method), 143–149 cube-root oscillator, 147 first-order harmonic balance, 145 methodology, 143–145 x ¨ + x1/3 = 0, 147–149 x ¨ + x3 = 0, 145–147 Harmonic balance (rational approximation), 149–153 Hamiltonian, 152 methodology, 149–153 rational procedure, 152

191 x¨ + x2 sgn(x) = 0, 152–153 x¨ + x3 = 0, 150–152 Harmonic oscillator, 128 Hyperbolic functions definition of, 14 dynamic system derivation, 17–18 elliptic functions as dynamic system, 40–41 |y|p − |x|q = 1, 119–126 I Iteration methods, 153–158 direct iteration scheme, 155–156 extended iteration, 157–158 inhomogeneous term, 154 J Jacobi elliptic differential equations, exact discretization of, 140–143 exact difference equation for cn, 141–142 exact difference equation for sn, 142–143 Jacobi cosine, 140 rescaled Duffing equation, 140 simple harmonic oscillator equation, 140 Jacobi elliptic functions addition theorems, 34 cn, sn, dn for special k values, 36 differential equations, 31–32 elementary properties of, 29 Fourier series, 36–37 fundamental relations for square of functions, 34 periodic solutions, 83, 86, 93 product relations, 36 special values of, 33–34

192 truly nonlinear oscillators, 133–137 Jacobi elliptic functions, dynamical system interpretation of, 37–40 bounds and symmetries, 39 definition of dynamic system, 37–38 discussion, 40 first integrals, 38–39 limits k → 0+ and k → 1− , 38 second-order differential equations, 40 N N-degrees-of-freedom (NDOF), 181 Newton’s force law, 128, 159 Null-clines, 10–13 O Odd function, 29, 52, 176 Oscillators antisymmetric, constant force, 134–135 cube-root, 147 Duffing, 128 harmonic, 128 Oscillators, truly nonlinear (TNL), 129, 133–137, 160 antisymmetric, constant force oscillator, 134–135 Jacobi elliptic function, 137 observations, 136 particle in a box, 136 restricted Duffing equation, 137 P Parabolic trigonometric functions, 67–81 definitions, 80 function names, 73, 80 geometric analysis of
|y| − 12 x2 = 12 78–80

Index geometric analysis of
|y| + 12 x2 =
12 72–73 H(x, y) = |y| + 21 x2 as a dynamic system, 67–72
|y| − 12 x2 = 12 as a dynamic system, 73–78 Particle in a box, 136 Periodic solutions, see also Generalized periodic solutions of f (t)2 + g(t)2 = 1 classes of, 165 Jacobi elliptic functions, 83, 86, 93 Product rules, 7–8 Pythagorean identity, 100 R Results obtained, 165–166 S Sine and cosine functions, trigonometric and hyperbolic, 1–22 addition and subtraction rules, 7 angular intervals, 20 derivatives, 5–6 discussion, 20–22 Euler relation, 1, 3 geometric proof that all trajectories are closed, 13 hyperbolic functions (dynamic system derivation), 17–18 hyperbolic sine and cosine (derived from sine and cosine), 14–17 integrals, 6 null-clines, 10–13 θ-periodic hyperbolic functions, 18–20 product rules, 7–8 sine and cosine (analytic definition), 3–8

Index sine and cosine (dynamic system approach), 8–14 sine and cosine (geometric definitions), 2–3 symmetry properties of trajectories in phase-space, 10 symmetry transformations, 11, 12 Taylor series, 6–7 unit circle, 2 x-y phase-space, 9–10 Special functions, 176–180 absolute value function, 177 delta function, 180 properties of even and odd functions, 176–177 sign and theta functions, 178 Square functions, 47–65 circular functions, 51 dynamic system interpretation of |x| + |y| = 1, 56–58 equations of motion, 56 Fourier series of square trigonometric functions, 53–55 hyperbolic square functions (dynamics system approach), 59–64 independent variable, 51 periodic hyperbolic square functions, 64–65 period of square trigonometric functions in the variable u(θ), 51–52 properties of square trigonometric functions, 50–51 Subtraction relations, 7

193 T Trigonometric hyperbolic relations, 174–176 basic properties, 174 derivatives and integrals, 174–175 exponential definition of hyperbolic functions, 174 related hyperbolic functions, 175 relations between hyperbolic and trigonometric functions, 175 sum and subtraction formulas, 175 Taylor series, 176 Trigonometric relations, 171–173 basic properties, 171 derivatives and integrals, 173 exponential definitions, 171 functions of sums of angles, 171 miscellaneous derived trigonometric functions, 173 miscellaneous trigonometric relations, 172 powers, 172 Taylor series, 173 Truly nonlinear (TNL) oscillators, 129, 133–137, 160 antisymmetric, constant force oscillator, 134–135 Jacobi elliptic function, 137 observations, 136 particle in a box, 136 restricted Duffing equation, 137 U Unresolved topics and issues, 166–168