Nonlinear Mechanics A Supplement to Theoretical Mechanics of Particles and Continua

by Fetter, Alexander L.; Walecka, John Dirk

• ISBN13: 9780486450315
• ISBN10: 0486450317
• Format: Paperback
• Copyright: 2006-06-16
• Publisher: Dover Publications

Summary

The authors' previous Dover book, Theoretical Mechanics of Particles and Continua,provided a lucid and self-contained account of classical mechanics, together with appropriate mathematical methods. This supplement and update of the original 1980 volume provides a bridge to contemporary mechanics. Alexander L. Fetter is Professor of Physics and Applied Physics at Stanford University, and John Dirk Walecka is Professor of Physics at the College of William and Mary. They are the authors of a prior Dover book, Quantum Theory of Many-Particle Systems.

Table of Contents

PART I: INTRODUCTION	1	(8)
1. Motivation	2	(7)
PART II: NONLINEAR CONTINUOUS SYSTEMS	9	(42)
2. Linearized stability analysis	10	(8)
Rayleigh-Taylor instability	10	(5)
Kelvin-Helmholtz instability	15	(3)
3. Rayleigh-Bénard problem: basic formulation	18	(8)
Boussinesq approximation and thermal expansion	20	(3)
Linearized perturbation equations	23	(1)
Boundary conditions	24	(2)
4. Rayleigh-Benard problem: linearized theory of convective instability	26	(11)
Proof that solutions are not oscillatory	26	(3)
Vorticity and the eigenvalue equation	29	(3)
Free-free boundary conditions: exact solution	32	(3)
Rigid-rigid boundary conditions: sketch of exact solution	35	(2)
5. Rayleigh-Bénard problem: expansion in Fourier modes	37	(8)
6. Lorenz equations: direct derivation for simple physical configuration	45	(6)
PART III: DISCRETE DYNAMICAL SYSTEMS	51	(85)
7. Example of a nonlinear oscillator	52	(6)
Duffing oscillator: general form	52	(2)
Duffing oscillator: perturbed simple harmonic oscillator	54	(4)
8. Phase-space dynamics and fixed points	58	(19)
Action-angle variables	61	(3)
Simple harmonic oscillator	64	(2)
Pendulum	66	(8)
Linearized stability analysis	74	(3)
9. Lorenz model	77	(10)
Stationary solutions and fixed points	78	(1)
Linearized stability analysis	79	(3)
Periodic oscillatory solutions	82	(2)
Chaotic solutions	84	(1)
Two theorems on phase-space convergence of solutions	85	(2)
10. Model finite-difference equation: logistic map	87	(10)
Logistic map	88	(1)
Power spectrum	89	(1)
Numerical analysis	90	(3)
Some analytic results	93	(4)
11. Liouville's theorem revisited	97	(4)
12. Action-angle variables revisited	101	(10)
Separable, periodic hamiltonian systems	101	(5)
Phase plots and motion on tori	106	(5)
13. Perturbation of periodic hamiltonian systems	111	(14)
Hamilton-Jacobi theory revisited	111	(4)
Direct perturbation analysis of the anharmonic oscillator	115	(2)
Model one-body problem with time-periodic perturbation	117	(3)
Resonant disruption of phase space	120	(5)
Overlap of disrupted regions	125	(1)
14. Coupled separable periodic hamiltonian systems	125	(11)
Two weakly coupled degrees of freedom	126	(3)
Resonant disruption of phase space	129	(5)
The Kolmogorov-Arnold-Moser (KAM) theorem	134	(2)
PART IV: PROBLEMS	136	(11)
15. Problems	137	(10)
REFERENCES	147	(3)
INDEX	150