Classical Mechanics: For Physics Graduate Students

by Corinaldesi, Ernesto

• ISBN13: 9789810236250
• ISBN10: 9810236255
• Format: Hardcover
• Copyright: 1999-06-01
• Publisher: World Scientific Pub Co Inc

Summary

This book is intended for first year physics graduate students who wish to learn about analytical mechanics. Lagrangians and Hamiltonians are extensively treated; particle motion, oscillations, coordinate systems, and rigid bodies are dealt with in far greater detail than in most undergraduate textbooks; and perturbation theory, relativistic mechanics, and two case studies of continuous systems are presented. Each subject is approached at progressively higher levels of abstraction. Lagrangians and Hamiltonians are first presented in an inductive way, leading up to general proofs. Hamiltonian mechanics is expressed in Cartan's notation not too early; there is a self-contained account of the traditional formulation.

Table of Contents

Introduction	1	(22)
Motion in phase space	1	(1)
Motion of a particle in one dimension	2	(3)
Flow in phase space	5	(3)
The action integral	8	(4)
The Maupertuis principle	12	(1)
The time	13	(3)
Fermat's principle	16	(2)
Chapter 1 problems	18	(5)
Examples of Particle Motion	23	(18)
Central forces	23	(2)
Circular and quasi-circular orbits	25	(1)
Isotropic harmonic oscillator	26	(1)
The Kepler problem	27	(4)
The L-R-L vector	31	(1)
Open Kepler-Rutherford orbits	32	(2)
Integrability	34	(2)
Chapter 2 problems	36	(5)
Fixed Points, Oscillations, Chaos	41	(26)
Fixed points	41	(3)
Small oscillations	44	(3)
Parametric resonance	47	(3)
Periodically jerked oscillator	50	(2)
Discrete maps, bifurcation, chaos	52	(4)
Chapter 3 problems	56	(11)
Coordinate Systems	67	(10)
Translations and rotations	67	(4)
Some kinematics	71	(1)
Fictitious forces	71	(2)
Chapter 4 problems	73	(4)
Rigid Bodies	77	(24)
Angular momentum	77	(1)
Euler's equations	78	(4)
Euler angles	82	(2)
Spinning top	84	(5)
Regular precession of top	85	(1)
Sleeping top	86	(1)
Irregular precessions of top	87	(2)
Gyrocompass	89	(2)
Tilted disk rolling in a circle	91	(2)
Chapter 5 problems	93	(8)
Lagrangians	101	(44)
Heuristic introduction	101	(2)
Velocity-dependent forces	103	(2)
Equivalent Lagrangians	105	(5)
Invariance of Lagrange equations	106	(2)
``Proofs'' of the Lagrange equations	108	(2)
Constraints	110	(5)
Invariance of L and constants of motion	115	(3)
Invariance of L up to total time derivative	118	(1)
Noether's theorem	119	(3)
Chapter 6 problem	122	(23)
Hamiltonians	145	(48)
First look. 1	145	(3)
First look. 2	148	(4)
H as Legendre transform of L	152	(2)
Liouville's theorem	154	(2)
Cartan's vectors and forms	156	(5)
Lie derivatives	161	(1)
Time-independent canonical transformations	162	(1)
Generating functions	163	(3)
Lagrange and Poisson brackets	166	(2)
Hamilton-Jacobi equation revisited	168	(2)
Time-dependent canonical transformations	170	(1)
Time-dependent Hamilton-Jacobi equation	171	(2)
Stokes' theorem and some proofs	173	(2)
Hamiltonian flow as a Lie-Cartan group	175	(3)
Poincare-Cartan integral invariant	178	(3)
Poincare's invariants	181	(1)
Chapter 7 problems	182	(11)
Action-Angle Variables	193	(20)
One dimension	193	(3)
Multiply periodic systems	196	(4)
Integrability, non-integrability, chaos	200	(2)
Adiabatic invariants	202	(2)
Outline of rigorous theory	204	(2)
Chapter 8 problems	206	(7)
Perturbation Theory	213	(14)
The operator Ω	213	(2)
Perturbation expansions	215	(2)
Perturbed periodic systems	217	(4)
Chapter 9 problems	221	(6)
Relativistic Dynamics	227	(34)
Lorentz transformations	227	(2)
Dynamics of a particle	229	(3)
Formulary of Lorentz transformations	232	(6)
The spinor connection	238	(3)
The spin	241	(3)
Thomas precession	244	(2)
Charged particle in static em field	246	(1)
Magnetic moment in static em field	247	(1)
Lagrangian and Hamiltonian	248	(5)
Chapter 10 problems	253	(8)
Continuous Systems	261	(22)
Uniform string	261	(7)
Ideal fluids	268	(7)
Chapter 11 problems	275	(8)
Index	283