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Introduction to Dynamics
BOOK
REVIEW
Ian Percival Cambridge
and Derek University
Richards,
Introduction
to Dynamics,
London,
Press, 1982, i +228 pp., $34.50, hardback
This book is an introduction to Hamiltonian dynamics designed to make this topic accessible to undergraduates. The reader is required to know nothing more than the theory of 2x2 matrices, ordinary differential equations, and the calculus of two variables. From this modest start, the authors introduce aspects of modem dynamics; they emphasize Hamiltonian systems with one degree of freedom and include other applications such as the dynamics of biological populations. The book culminates with a brief discussion of the theory of chaos in nonlinear maps. The book consistently introduces new concepts by the simplest means possible and elaborates these concepts by examples and exercises. The first three chapters use this technique to lay the requisite groundwork of first- and secondorder autonomous systems and to emphasize the classification of stationary points and the geometry of associated phase curves. With this background, the authors proceed to the theory of Hamiltonian systems with one degree of freedom. After giving several concrete phase diagrams, they develop general formulas for system periods and prove Liouville’s Theorem for time-dependent systems. The Lagrange equations of motion and a particularly lucid treatment of canonical transformations follow. The theory of angle-action variables completes the basic study of Hamiltonian systems with one degree of freedom. The next chapter provides an elementary introduction to perturbation techniques and gives first-order perturbation theory for conservative Hamiltonian systems. The perturbation theory is then applied to periodic systems with a time-dependent Hamiltonian that varies slowly or oscillates rapidly. The final two chapters are devoted to chaos. Propagators of time-dependent linear systems serve as an introduction to nonlinear systems. The logistic equation gives an example of one-dimensional chaos, and the Cremona map of Siegel and Henon is then discussed. The aim of this book clearly has been realized: concepts in dynamics are introduced by example, with a minimum of notational baggage. Because of this, this book should be a welcome adjunct to texts that give the full multidimensional generality of dynamical systems.
MATHEMATICAL
BIOSCIENCES
75:279-280
OElsevier Science Publishing Co., Inc., 1985 52 Vanderbilt Ave., New York, NY 10017
(1985)
279 0025-5564/85/$03.30
280
BOOK
REVIEW
In conclusion, this book is an excellent introduction to Hamiltonian systems and the techniques for solving them. While aimed primarily at the undergraduate level, it is recommended as an introduction to dynamical systems for graduates and research workers as well. JOHN L. SPOUGE Theoretical Biology and Bioph.vsics Theoretical Division Los Alamos National Laboratory Los Alamos, NM 87545
REVIEW
Ian Percival Cambridge
and Derek University
Richards,
Introduction
to Dynamics,
London,
Press, 1982, i +228 pp., $34.50, hardback
This book is an introduction to Hamiltonian dynamics designed to make this topic accessible to undergraduates. The reader is required to know nothing more than the theory of 2x2 matrices, ordinary differential equations, and the calculus of two variables. From this modest start, the authors introduce aspects of modem dynamics; they emphasize Hamiltonian systems with one degree of freedom and include other applications such as the dynamics of biological populations. The book culminates with a brief discussion of the theory of chaos in nonlinear maps. The book consistently introduces new concepts by the simplest means possible and elaborates these concepts by examples and exercises. The first three chapters use this technique to lay the requisite groundwork of first- and secondorder autonomous systems and to emphasize the classification of stationary points and the geometry of associated phase curves. With this background, the authors proceed to the theory of Hamiltonian systems with one degree of freedom. After giving several concrete phase diagrams, they develop general formulas for system periods and prove Liouville’s Theorem for time-dependent systems. The Lagrange equations of motion and a particularly lucid treatment of canonical transformations follow. The theory of angle-action variables completes the basic study of Hamiltonian systems with one degree of freedom. The next chapter provides an elementary introduction to perturbation techniques and gives first-order perturbation theory for conservative Hamiltonian systems. The perturbation theory is then applied to periodic systems with a time-dependent Hamiltonian that varies slowly or oscillates rapidly. The final two chapters are devoted to chaos. Propagators of time-dependent linear systems serve as an introduction to nonlinear systems. The logistic equation gives an example of one-dimensional chaos, and the Cremona map of Siegel and Henon is then discussed. The aim of this book clearly has been realized: concepts in dynamics are introduced by example, with a minimum of notational baggage. Because of this, this book should be a welcome adjunct to texts that give the full multidimensional generality of dynamical systems.
MATHEMATICAL
BIOSCIENCES
75:279-280
OElsevier Science Publishing Co., Inc., 1985 52 Vanderbilt Ave., New York, NY 10017
(1985)
279 0025-5564/85/$03.30
280
BOOK
REVIEW
In conclusion, this book is an excellent introduction to Hamiltonian systems and the techniques for solving them. While aimed primarily at the undergraduate level, it is recommended as an introduction to dynamical systems for graduates and research workers as well. JOHN L. SPOUGE Theoretical Biology and Bioph.vsics Theoretical Division Los Alamos National Laboratory Los Alamos, NM 87545