Classical Mechanics

by Kibble, T. W. B.; Berkshire, Frank H.; Berkshire, F. H.

• ISBN13: 9781860944246
• ISBN10: 1860944248
• Edition: 5th
• Format: Hardcover
• Copyright: 2004-06-03
• Publisher: Imperial College Pr

Summary

This is the fifth edition of a well-established textbook. It is intended to provide a thorough coverage of the fundamental principles and techniques of classical mechanics, an old subject that is at the base of all of physics, but in which there has also in recent years been rapid development. It emphasizes the basic principles, and aims to progress rapidly to the point of being able to handle physically and mathematically interesting problems, without getting bogged down in excessive formalism. Lagrangian methods are introduced at a relatively early stage, to get students to appreciate their use in simple contexts. Later chapters use Lagrangian and Hamiltonian methods extensively, but in a way that aims to be accessible to under-graduates, while including modern develop-ments at the appropriate level of detail. The subject has been developed considerably recently while retaining a truly central role for all students of physics and applied mathematics.

Table of Contents

Preface	vii
Useful Constants and Units	xv
List of Symbols	xvii
Introduction	1	(16)
Space and Time	2	(3)
Newton's Laws	5	(5)
The Concepts of Mass and Force	10	(3)
External Forces	13	(1)
Summary	13	(4)
Linear Motion	17	(32)
Conservative Forces; Conservation of Energy	17	(3)
Motion near Equilibrium; the Harmonic Oscillator	20	(4)
Complex Representation	24	(1)
The Law of Conservation of Energy	25	(2)
The Damped Oscillator	27	(3)
Oscillator under Simple Periodic Force	30	(4)
General Periodic Force	34	(3)
Impulsive Forces; the Green's Function Method	37	(2)
Collision Problems	39	(3)
Summary	42	(7)
Energy and Angular Momentum	49	(24)
Energy; Conservative Forces	49	(2)
Projectiles	51	(2)
Moments; Angular Momentum	53	(2)
Central Forces; Conservation of Angular Momentum	55	(2)
Polar Co-ordinates	57	(2)
The Calculus of Variations	59	(3)
Hamilton's Principle; Lagrange's Equations	62	(4)
Summary	66	(7)
Central Conservative Forces	73	(32)
The Isotropic Harmonic Oscillator	73	(3)
The Conservation Laws	76	(2)
The Inverse Square Law	78	(6)
Orbits	84	(6)
Scattering Cross-sections	90	(4)
Mean Free Path	94	(2)
Rutherford Scattering	96	(2)
Summary	98	(7)
Rotating Frames	105	(24)
Angular Velocity; Rate of Change of a Vector	105	(3)
Particle in a Uniform Magnetic Field	108	(3)
Acceleration; Apparent Gravity	111	(3)
Coriolis Force	114	(6)
Larmor Effect	120	(1)
Angular Momentum and the Larmor Effect	121	(3)
Summary	124	(5)
Potential Theory	129	(30)
Gravitational and Electrostatic Potentials	129	(2)
The Dipole and Quadrupole	131	(3)
Spherical Charge Distributions	134	(3)
Expansion of Potential at Large Distances	137	(3)
The Shape of the Earth	140	(4)
The Tides	144	(4)
The Field Equations	148	(4)
Summary	152	(7)
The Two-Body Problem	159	(18)
Centre-of-mass and Relative Co-ordinates	159	(3)
The Centre-of-mass Frame	162	(3)
Elastic Collisions	165	(3)
CM and Lab Cross-sections	168	(5)
Summary	173	(4)
Many-Body Systems	177	(20)
Momentum; Centre-of-mass Motion	177	(4)
Angular Momentum; Central Internal Forces	181	(2)
The Earth--Moon System	183	(5)
Energy; Conservative Forces	188	(2)
Lagrange's Equations	190	(2)
Summary	192	(5)
Rigid Bodies	197	(34)
Basic Principles	197	(1)
Rotation about an Axis	198	(5)
Perpendicular Components of Angular Momentum	203	(2)
Principal Axes of Inertia	205	(3)
Calculation of Moments of Inertia	208	(3)
Effect of a Small Force on the Axis	211	(5)
Instantaneous Angular Velocity	216	(2)
Rotation about a Principal Axis	218	(3)
Euler's Angles	221	(4)
Summary	225	(6)
Lagrangian Mechanics	231	(22)
Generalized Co-ordinates; Holonomic Systems	231	(2)
Lagrange's Equations	233	(3)
Precession of a Symmetric Top	236	(2)
Pendulum Constrained to Rotate about an Axis	238	(3)
Charged Particle in an Electromagnetic Field	241	(3)
The Stretched String	244	(4)
Summary	248	(5)
Small Oscillations and Normal Modes	253	(24)
Orthogonal Co-ordinates	253	(3)
Equations of Motion for Small Oscillations	256	(2)
Normal Modes	258	(3)
Coupled Oscillators	261	(5)
Oscillations of Particles on a String	266	(3)
Normal Modes of a Stretched String	269	(3)
Summary	272	(5)
Hamiltonian Mechanics	277	(30)
Hamilton's Equations	277	(3)
Conservation of Energy	280	(2)
Ignorable Co-ordinates	282	(3)
General Motion of the Symmetric Top	285	(4)
Liouville's Theorem	289	(2)
Symmetries and Conservation Laws	291	(4)
Galilean Transformations	295	(5)
Summary	300	(7)
Dynamical Systems and Their Geometry	307	(40)
Phase Space and Phase Portraits	307	(2)
First-order Systems --- the Phase Line (n = 1)	309	(3)
Second-order Systems --- the Phase Plane (n = 2)	312	(6)
Prey--Predator, Competing-species Systems and War	318	(6)
Limit Cycles	324	(5)
Systems of Third (and Higher) Order	329	(8)
Sensitivity to Initial Conditions and Predictability	337	(3)
Summary	340	(7)
Order and Chaos in Hamiltonian Systems	347	(34)
Integrability	347	(4)
Surfaces of Section	351	(3)
Action/Angle Variables	354	(5)
Some Hamiltonian Systems which Exhibit Chaos	359	(10)
Slow Change of Parameters --- Adiabatic Invariance	369	(3)
Near-integrable Systems	372	(2)
Summary	374	(7)
Appendix A. Vectors	381	(28)
A.1 Definitions and Elementary Properties	381	(3)
A.2 The Scalar Product	384	(1)
A.3 The Vector Product	385	(3)
A.4 Differentiation and Integration of Vectors	388	(2)
A.5 Gradient, Divergence and Curl	390	(3)
A.6 Integral Theorems	393	(4)
A.7 Electromagnetic Potentials	397	(1)
A.8 Curvilinear Co-ordinates	398	(3)
A.9 Tensors	401	(2)
A.10 Eigenvalues; Diagonalization of a Symmetric Tensor	403	(6)
Appendix B. Conics	409	(6)
B.1 Cartesian Form	409	(3)
B.2 Polar Form	412	(3)
Appendix C. Phase Plane Analysis near Critical Points	415	(30)
C.1 Linear Systems and their Classification	415	(6)
C.2 Almost Linear Systems	421	(2)
C.3 Systems of Third (and Higher) Order	423	(2)
Appendix D. Discrete Dynamical Systems --- Maps	425	(20)
D.1 One-dimensional Maps	425	(8)
D.2 Two-dimensional Maps	433	(4)
D.3 Twist Maps and Torus Breakdown	437	(8)
Answers to Problems	445	(18)
Bibliography	463	(2)
Index	465