Lagrangian and Hamiltonian Mechanics

by Calkin, M. G.

• ISBN13: 9789810226725
• ISBN10: 9810226721
• Format: Hardcover
• Copyright: 1996-06-01
• Publisher: World Scientific Pub Co Inc

Table of Contents

Preface	v
CHAPTER I NEWTON'S LAWS	1	(27)
Newton's laws	1	(2)
Free fall	3	(1)
Simple harmonic oscillator	4	(2)
Central force	6	(3)
Gravitational force: qualitative	9	(2)
Gravitational force: quantitative	11	(5)
Parameters of earth's orbit	16	(3)
Scattering	19	(2)
Coulomb scattering	21	(1)
Exercises	22	(6)
CHAPTER II THE PRINCIPLE OF VIRTUAL WORK AND D'ALEMBERT'S PRINCIPLE	28	(12)
Constraints	28	(1)
Principle of virtual work	29	(3)
D'Alembert's principle and generalized coordinates	32	(1)
Lever	33	(1)
Inclined plane	34	(1)
Plane pendulum	34	(3)
Exercises	37	(3)
CHAPTER III LAGRANGE'S EQUATIONS	40	(19)
Lagrange's equations	40	(4)
Plane pendulum	44	(1)
Spherical pendulum	45	(1)
Electromagnetic interaction	46	(3)
Interaction of an electric charge and a magnet	49	(5)
Exercises	54	(5)
CHAPTER IV THE PRINCIPLE OF STATIONARY ACTION OR HAMILTON'S PRINCIPLE	59	(21)
Principle of stationary action	59	(3)
Calculus of variations	62	(2)
Geodesics	64	(7)
Examples	67	(4)
Path integral formulation of quantum mechanics	71	(4)
Exercises	75	(5)
CHAPTER V INVARIANCE TRANSFORMATIONS AND CONSTANTS OF THE MOTION	80	(18)
Invariance transformations	80	(1)
Free particle (a)	81	(1)
Infinitesimal transformations	82	(1)
Free particle (b)	83	(1)
Space time transformations	84	(6)
Spatial displacement	85	(1)
Spatial rotation	85	(2)
Galilean transformation	87	(1)
Time displacement	88	(2)
Covariance, invariance, and the action	90	(6)
Exercises	96	(2)
CHAPTER VI HAMILTON'S EQUATIONS	98	(17)
Hamilton's equations	98	(3)
Plane pendulum	101	(2)
Spherical pendulum	103	(1)
Rotating pendulum	104	(3)
Electromagnetic interaction	107	(1)
Poisson brackets	107	(3)
Exercises	110	(5)
CHAPTER VII CANONICAL TRANSFORMATIONS	115	(28)
One degree of freedom	115	(4)
Generating functions	119	(4)
Identity and point transformations	123	(2)
Infinitesimal canonical transformations	125	(1)
Invariance transformations	126	(2)
Lagrange and Poisson brackets	128	(4)
Time dependence	132	(2)
Integral invariants	134	(6)
Exercises	140	(3)
CHAPTER VIII HAMILTON-JACOBI THEORY	143	(28)
Hamilton's principal function	143	(4)
Jacobi's complete integral	147	(4)
Time-independent Hamilton-Jacobi equation	151	(4)
Separation of variables	155	(1)
Free particle, in cartesian coordinates	155	(2)
Central force, in spherical polar coordinates	157	(5)
Hamilton-Jacobi mechanics, geometric optics, and wave mechanics	162	(5)
Exercises	167	(4)
CHAPTER IX ACTION-ANGLE VARIABLES	171	(20)
Action-angle variables	171	(5)
Example: simple harmonic oscillator	176	(2)
Example: central force	178	(3)
Adiabatic change	181	(6)
Exercises	187	(4)
CHAPTER X NON-INTEGRABLE SYSTEMS	191	(19)
Surface of section	191	(5)
Integrable and non-integrable systems	196	(3)
Perturbation theory	199	(2)
Irrational tori	201	(2)
Rational tori	203	(4)
Exercises	207	(3)
INDEX	210