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Introduction to Dynamics - Ian C. Percival

Paperback Published: 31st January 1983
ISBN: 9780521281492
Number Of Pages: 240

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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.

'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

Prefacep. viii
First-order autonomous systemsp. 1
Basic theoryp. 1
Rotationp. 6
Natural boundariesp. 7
Examples from biologyp. 8
Exercisesp. 9
Linear transformations of the planep. 12
Introductionp. 12
Area-preserving transformationsp. 13
Transformations with dilationp. 18
Exercisesp. 21
Second-order autonomous systemsp. 23
Systems of order np. 23
Phase flows of second-order autonomous systemsp. 25
Fixed points, equilibrium and stabilityp. 26
Separation of variablesp. 28
Classification of fixed pointsp. 32
Summary of classificationp. 34
Determination of fixed pointsp. 36
Limit cyclesp. 37
Exercisesp. 38
Conservative Hamiltonian systems of one degree of freedomp. 42
Newtonian and Hamiltonian systemsp. 42
Conservative systemsp. 44
Linear conservative systemsp. 45
The cubic potentialp. 49
General potentialp. 51
Free rotationsp. 53
The vertical pendulump. 54
Rotation, libration and periodsp. 56
Area-preserving flows and Liouville's theoremp. 57
Exercisesp. 60
Lagrangiansp. 66
Introductionp. 66
The Legendre transformationp. 67
The Lagrangian equation of motionp. 70
Formulationp. 72
Exercisesp. 78
Transformation theoryp. 84
Introductionp. 84
The theory of time-independent transformationsp. 84
The F[subscript 1] (Q, q) generating functionp. 87
Other forms of generating functionp. 89
The transformed Hamiltonianp. 92
Time-dependent transformationsp. 94
Hamiltonians under time-dependent transformationsp. 96
Group property and infinitesimal canonical transformationsp. 97
Exercisesp. 99
Angle-action variablesp. 103
The simplest variablesp. 103
The Hamiltonian in angle-action representationp. 106
The dependence of the angle variable upon qp. 110
Generating functionsp. 112
Rotationsp. 114
Exercisesp. 116
Perturbation theoryp. 122
Introductionp. 122
First-order perturbation theory for conservative Hamiltonian systemsp. 125
Exercisesp. 133
Adiabatic and rapidly oscillating conditionsp. 141
Introductionp. 141
Elastic ball bouncing between two slowly moving planesp. 142
The linear oscillator with a slowly changing frequencyp. 144
General adiabatic theoryp. 149
Motion in a rapidly oscillating field: fast perturbationsp. 153
Exercisesp. 157
Linear Systemsp. 163
Introductionp. 163
First-order systemsp. 163
Forced linear oscillatorp. 170
Propagatorsp. 173
Periodic conditions and linear mapsp. 179
Linear area-preserving mapsp. 181
Periodic forces and parametric resonancep. 186
Exercisesp. 191
Chaotic motion and non-linear mapsp. 197
Chaotic motionp. 197
Maps and discrete timep. 198
The logistic mapp. 199
Quadratic area-preserving mapsp. 205
Regular and chaotic motion of Hamiltonian systemsp. 213
Exercisesp. 214
Existence theoremsp. 216
Integrals required for some soluble problemsp. 219
Indexp. 223
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521281492
ISBN-10: 0521281490
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 240
Published: 31st January 1983
Country of Publication: GB
Dimensions (cm): 22.96 x 15.29  x 1.52
Weight (kg): 0.37

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