Introduction to Dynamics

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Paperback | 31 January 1983

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240 Pages

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Recent advances in dynamics, with wide applications throughout the sciences and engineering, have meant that a new approach to the subject is needed. Furthermore, the mathematical and scientific background of students has changed in recent years. In this book, the subject of dynamics is introduced at undergraduate level through the elementary qualitative theory of differential equations, the geometry of phase curves and the theory of stability. Each subject, from the most elementary topic to some important discoveries of recent decades, is introduced through simple examples and illustrated with many diagrams. The text is supplemented with over a hundred exercises. The examples and exercises cover subjects as diverse as mechanics and population dynamics.

The mathematical background required of the reader is an understanding of the elementary theory of differential equations and matrix arithmetic. The book will be of interest to second-year and third-year undergraduates at universities, polytechnics and technical colleges who are studying science and engineering courses. It is also suitable for graduates and research workers in such fields as plasma, atomic, particle and molecular physics, astronomy and theoretical ecology.

Industry Reviews

'Percival and Richards give a beautifully clear introduction to dynamics, discussing why scientists should be interested in the stability properties of their equations, and explaining the terminology.' Nature 'The authors of this splendid textbook have made a commendable effort to introduce the geometric and qualitative aspects of dynamics at an undergraduate level ... an exciting, well-organised book with many illuminating worked examples. A course based on this book would be a pleasure to teach. It should be greatly appreciated by physicists and engineers, and might help bring applied mathematics forward into the nineteenth century.' Times Higher Education Supplement

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Preface	p. viii
First-order autonomous systems	p. 1
Basic theory	p. 1
Rotation	p. 6
Natural boundaries	p. 7
Examples from biology	p. 8
Exercises	p. 9
Linear transformations of the plane	p. 12
Introduction	p. 12
Area-preserving transformations	p. 13
Transformations with dilation	p. 18
Exercises	p. 21
Second-order autonomous systems	p. 23
Systems of order n	p. 23
Phase flows of second-order autonomous systems	p. 25
Fixed points, equilibrium and stability	p. 26
Separation of variables	p. 28
Classification of fixed points	p. 32
Summary of classification	p. 34
Determination of fixed points	p. 36
Limit cycles	p. 37
Exercises	p. 38
Conservative Hamiltonian systems of one degree of freedom	p. 42
Newtonian and Hamiltonian systems	p. 42
Conservative systems	p. 44
Linear conservative systems	p. 45
The cubic potential	p. 49
General potential	p. 51
Free rotations	p. 53
The vertical pendulum	p. 54
Rotation, libration and periods	p. 56
Area-preserving flows and Liouville's theorem	p. 57
Exercises	p. 60
Lagrangians	p. 66
Introduction	p. 66
The Legendre transformation	p. 67
The Lagrangian equation of motion	p. 70
Formulation	p. 72
Exercises	p. 78
Transformation theory	p. 84
Introduction	p. 84
The theory of time-independent transformations	p. 84
The F[subscript 1] (Q, q) generating function	p. 87
Other forms of generating function	p. 89
The transformed Hamiltonian	p. 92
Time-dependent transformations	p. 94
Hamiltonians under time-dependent transformations	p. 96
Group property and infinitesimal canonical transformations	p. 97
Exercises	p. 99
Angle-action variables	p. 103
The simplest variables	p. 103
The Hamiltonian in angle-action representation	p. 106
The dependence of the angle variable upon q	p. 110
Generating functions	p. 112
Rotations	p. 114
Exercises	p. 116
Perturbation theory	p. 122
Introduction	p. 122
First-order perturbation theory for conservative Hamiltonian systems	p. 125
Exercises	p. 133
Adiabatic and rapidly oscillating conditions	p. 141
Introduction	p. 141
Elastic ball bouncing between two slowly moving planes	p. 142
The linear oscillator with a slowly changing frequency	p. 144
General adiabatic theory	p. 149
Motion in a rapidly oscillating field: fast perturbations	p. 153
Exercises	p. 157
Linear Systems	p. 163
Introduction	p. 163
First-order systems	p. 163
Forced linear oscillator	p. 170
Propagators	p. 173
Periodic conditions and linear maps	p. 179
Linear area-preserving maps	p. 181
Periodic forces and parametric resonance	p. 186
Exercises	p. 191
Chaotic motion and non-linear maps	p. 197
Chaotic motion	p. 197
Maps and discrete time	p. 198
The logistic map	p. 199
Quadratic area-preserving maps	p. 205
Regular and chaotic motion of Hamiltonian systems	p. 213
Exercises	p. 214
Existence theorems	p. 216
Integrals required for some soluble problems	p. 219
Index	p. 223
Table of Contents provided by Syndetics. All Rights Reserved.

Introduction to Dynamics

At a Glance

Paperback

Industry Reviews

Shipping

How to return your order

Defective items

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