Classical Mechanics

by R. Douglas Gregory

• ISBN13: 9780521826785
• ISBN10: 0521826780
• Format: Hardcover
• Copyright: 2006-04-24
• Publisher: Cambridge University Press

Summary

Mechanics is the study of the motion of physical objects. As such, the subject finds its applications in all areas of physics and engineering. Topics covered in this complete guide to mechanics include Newton's, Lagrange's and Hamilton's equations of motion, kinematics, oscillation, particle mechanics and rigid body motion. A thorough understanding of the theory is provided by illustrating the classical laws of motion with many relevant examples, Matlab code and case studies. Solutions to exercises are available.

Table of Contents

Preface	xi
Newtonian mechanics of a single particle	1	(218)
The algebra and calculus of vectors	3	(22)
Vectors and vector quantities	3	(2)
Linear operations: a + b and λa	5	(5)
The scalar product a . b	10	(3)
The vector product a x b	13	(2)
Triple products	15	(1)
Vector functions of a scalar variable	16	(2)
Tangent and normal vectors to a curve	18	(7)
Problems	22	(3)
Velocity, acceleration and scalar angular velocity	25	(25)
Straight line motion of a particle	25	(3)
General motion of a particle	28	(4)
Particle motion in polar co-ordinates	32	(4)
Rigid body rotating about a fixed axis	36	(2)
Rigid body in planar motion	38	(2)
Reference frames in relative motion	40	(10)
Problems	43	(7)
Newton's laws of motion and the law of gravitation	50	(23)
Newton's laws of motion	50	(2)
Inertial frames and the law of inertia	52	(2)
The law of mutual interaction; mass and force	54	(3)
The law of multiple interactions	57	(1)
Centre of mass	58	(1)
The law of gravitation	59	(1)
Gravitation by a distribution of mass	60	(7)
The principle of equivalence and g	67	(6)
Problems	71	(2)
Problems in particle dynamics	73	(32)
Rectilinear motion in a force field	74	(4)
Constrained rectilinear motion	78	(4)
Motion through a resisting medium	82	(6)
Projectiles	88	(4)
Circular motion	92	(13)
Problems	98	(7)
Linear oscillations	105	(26)
Body on a spring	105	(2)
Classical simple harmonic motion	107	(2)
Damped simple harmonic motion	109	(3)
Driven (forced) motion	112	(8)
A simple seismograph	120	(1)
Coupled oscillations and normal modes	121	(10)
Problems	126	(5)
Energy conservation	131	(24)
The energy principle	131	(2)
Energy conservation in rectilinear motion	133	(3)
General features of rectilinear motion	136	(4)
Energy conservation in a conservative field	140	(5)
Energy conservation in constrained motion	145	(10)
Problems	151	(4)
Orbits in a central field	155	(39)
The one-body problem -- Newton's equations	157	(2)
General nature of orbital motion	159	(5)
The path equation	164	(3)
Nearly circular orbits	167	(3)
The attractive inverse square field	170	(7)
Space travel -- Hohmann transfer orbits	177	(2)
The repulsive inverse square field	179	(1)
Rutherford scattering	179	(15)
Appendix A The geometry of conics	184	(2)
Appendix B The Hohmann orbit is optimal	186	(2)
Problems	188	(6)
Non-linear oscillations and phase space	194	(25)
Periodic non-linear oscillations	194	(5)
The phase plane ((x1, x2)--plane)	199	(3)
The phase plane in dynamics ((x, v)--plane)	202	(3)
Poincare-Bendixson theorem: limit cycles	205	(6)
Driven non-linear oscillations	211	(8)
Problems	214	(5)
Multi-particle systems	219	(102)
The energy principle	221	(24)
Configurations and degrees of freedom	221	(2)
The energy principle for a system	223	(2)
Energy conservation for a system	225	(8)
Kinetic energy of a rigid body	233	(12)
Problems	241	(4)
The linear momentum principle	245	(41)
Linear momentum	245	(1)
The linear momentum principle	246	(1)
Motion of the centre of mass	247	(3)
Conservation of linear momentum	250	(1)
Rocket motion	251	(4)
Collision theory	255	(4)
Collision processes in the zero-momentum frame	259	(5)
The two-body problem	264	(5)
Two-body scattering	269	(4)
Integrable mechanical systems	273	(13)
Appendix A Modelling bodies by particles	277	(2)
Problems	279	(7)
The angular momentum principle	286	(35)
The moment of a force	286	(3)
Angular momentum	289	(3)
Angular momentum of a rigid body	292	(2)
The angular momentum principle	294	(4)
Conservation of angular momentum	298	(8)
Planar rigid body motion	306	(7)
Rigid body statics in three dimensions	313	(8)
Problems	317	(4)
Analytical mechanics	321	(98)
Lagrange's equations and conservation principles	323	(43)
Constraints and constraint forces	323	(2)
Generalised coordinates	325	(5)
Configuration space (q--space)	330	(3)
D'Alembert's principle	333	(2)
Lagrange's equations	335	(9)
Systems with moving constraints	344	(4)
The Lagrangian	348	(3)
The energy function h	351	(3)
Generalised momenta	354	(2)
Symmetry and conservation principles	356	(10)
Problems	361	(5)
The calculus of variations and Hamilton's principle	366	(27)
Some typical minimisation problems	367	(2)
The Euler--Lagrange equation	369	(11)
Variational principles	380	(3)
Hamilton's principle	383	(10)
Problems	388	(5)
Hamilton's equations and phase space	393	(26)
Systems of first order ODEs	393	(3)
Legendre transforms	396	(4)
Hamilton's equations	400	(6)
Hamiltonian phase space ((q, p)--space)	406	(2)
Liouville's theorem and recurrence	408	(11)
Problems	413	(6)
Further topics	419	(145)
The general theory of small oscillations	421	(36)
Stable equilibrium and small oscillations	421	(4)
The approximate forms of T and V	425	(4)
The general theory of normal modes	429	(4)
Existence theory for normal modes	433	(3)
Some typical normal mode problems	436	(8)
Orthogonality of normal modes	444	(3)
General small oscillations	447	(1)
Normal coordinates	448	(9)
Problems	452	(5)
Vector angular velocity and rigid body kinematics	457	(12)
Rotation about a fixed axis	457	(3)
General rigid body kinematics	460	(9)
Problems	467	(2)
Rotating reference frames	469	(23)
Transformation formulae	469	(7)
Particle dynamics in a non-inertial frame	476	(2)
Motion relative to the Earth	478	(7)
Multi-particle system in a non-inertial frame	485	(7)
Problems	489	(3)
Tensor algebra and the inertia tensor	492	(30)
Orthogonal transformations	493	(2)
Rotated and reflected coordinate systems	495	(4)
Scalars, vectors and tensors	499	(6)
Tensor algebra	505	(3)
The inertia tensor	508	(6)
Principal axes of a symmetric tensor	514	(2)
Dynamical symmetry	516	(6)
Problems	519	(3)
Problems in rigid body dynamics	522	(42)
Equations of rigid body dynamics	522	(2)
Motion of `spheres'	524	(1)
The snooker ball	525	(2)
Free motion of bodies with axial symmetry	527	(4)
The spinning top	531	(4)
Lagrangian dynamics of the top	535	(6)
The gyrocompass	541	(3)
Euler's equations	544	(5)
Free motion of an unsymmetrical body	549	(7)
The rolling wheel	556	(8)
Problems	560	(4)
Appendix Centres of mass and moments of inertia	564	(12)
Centre of mass	564	(3)
Moment of inertia	567	(4)
Parallel and perpendicular axes	571	(5)
Answers to the problems	576	(13)
Bibliography	589	(2)
Index	591