Classical Dynamics of Particles and Systems

by Thornton, Stephen T.; Marion, Jerry B.

• ISBN13: 9780030973024
• ISBN10: 0030973023
• Edition: 4th
• Format: Hardcover
• Copyright: 1995-01-17
• Publisher: Brooks Cole

Summary

This best-selling classical mechanics text, written for the advanced undergraduate one- or two-semester course, provides a complete account of the classical mechanics of particles, systems of particles, and rigid bodies. The authors make extensive use of vector calculus to explore topics; coverage also includes the Lagrangian formulation of mechanics. Modern notation and terminology are used throughout in support of the text's objective: to facilitate the transition to the quantum theory of physics.

Table of Contents

1 MATRICES, VECTORS, AND VECTOR CALCULUS	1	(47)
1.1 Introduction	1	(1)
1.2 Concept of a Scalar	2	(1)
1.3 Coordinate Transformations	3	(3)
1.4 Properties of Rotation Matrices	6	(3)
1.5 Matrix Operations	9	(3)
1.6 Further Definitions	12	(2)
1.7 Geometrical Significance of Transformation Matrices	14	(6)
1.8 Definitions of a Scalar and a Vector in Terms of Transformation Properties	20	(1)
1.9 Elementary Scalar and Vector Operations	21	(1)
1.10 Scalar Product of Two Vectors	21	(3)
1.11 Unit Vectors	24	(1)
1.12 Vector Product of Two Vectors	25	(5)
1.13 Differentiation of a Vector with Respect to a Scalar	30	(1)
1.14 Examples of Derivatives-Velocity and Acceleration	31	(4)
1.15 Angular Velocity	35	(3)
1.16 Gradient Operator	38	(3)
1.17 Integration of Vectors	41	(3)
Problems	44	(4)
2 NEWTONIAN MECHANICS-SINGLE PARTICLE	48	(59)
2.1 Introduction	48	(1)
2.2 Newton's Laws	49	(4)
2.3 Frames of Reference	53	(1)
2.4 The Equation of Motion for a Particle	54	(22)
2.5 Conservation Theorems	76	(6)
2.6 Energy	82	(6)
2.7 Rocket Motion	88	(9)
2.8 Limitations of Newtonian Mechanics	97	(2)
Problems	99	(8)
3 OSCILLATIONS	107	(46)
3.1 Introduction	107	(1)
3.2 Simple Harmonic Oscillator	108	(3)
3.3 Harmonic Oscillations in Two Dimensions	111	(3)
3.4 Phase Diagrams	114	(2)
3.5 Damped Oscillations	116	(9)
3.6 Sinusoidal Driving Forces	125	(6)
3.7 Physical Systems	131	(1)
3.8 Electrical Oscillations	132	(5)
3.9 Principle of Superposition-Fourier Series	137	(3)
3.10 The Response of Linear Oscillators to Impulsive Forcing Functions (optional)	140	(8)
Problems	148	(5)
4 NONLINEAR OSCILLATIONS AND CHAOS	153	(36)
4.1 Introduction	153	(2)
4.2 Nonlinear Oscillations	155	(4)
4.3 Phase Diagrams for Nonlinear Systems	159	(3)
4.4 Plane Pendulum	162	(5)
4.5 Jumps, Hysteresis, and Phase Lags	167	(4)
4.6 Chaos in a Pendulum	171	(6)
4.7 Mapping	177	(4)
4.8 Chaos Identification	181	(5)
Problems	186	(3)
5 GRAVITATION	189	(24)
5.1 Introduction	189	(2)
5.2 Gravitational Potential	191	(9)
5.3 Lines of Force and Equipotential Surfaces	200	(1)
5.4 When Is the Potential Concept Useful?	201	(3)
5.5 Ocean Tides	204	(6)
Problems	210	(3)
6 SOME METHODS IN THE CALCULUS OF VARIATIONS	213	(19)
6.1 Introduction	213	(1)
6.2 Statement of the Problem	214	(2)
6.3 Euler's Equation	216	(6)
6.4 The "Second Form" of the Euler Equation	222	(3)
6.5 Functions with Several Dependent Variables	225	(1)
6.6 Euler Equations When Auxiliary Conditions Are Imposed	225	(3)
6.7 The (XXX) Notation	228	(2)
Problems	230	(2)
7 HAMILTON'S PRINCIPLE-LAGRANGIAN AND HAMILTONIAN DYNAMICS	232	(59)
7.1 Introduction	232	(1)
7.2 Hamilton's Principle	233	(4)
7.3 Generalized Coordinates	237	(4)
7.4 Lagrange's Equations of Motion in Generalized Coordinates	241	(11)
7.5 Lagrange's Equations with Undetermined Multipliers	252	(6)
7.6 Equivalence of Lagrange's and Newton's Equations	258	(3)
7.7 Essence of Lagrangian Dynamics	261	(2)
7.8 A Theorem Concerning the Kinetic Energy	263	(1)
7.9 Conservation Theorems Revisited	264	(5)
7.10 Canonical Equations of Motion-Hamiltonian Dynamics	269	(8)
7.11 Some Comments Regarding Dynamical Variables and Variational Calculations in Physics	277	(2)
7.12 Phase Space and Liouville's Theorem (optional)	279	(3)
7.13 Virial Theorem (optional)	282	(3)
Problems	285	(6)
8 CENTRAL-FORCE MOTION	291	(42)
8.1 Introduction	291	(1)
8.2 Reduced Mass	291	(2)
8.3 Conservation Theorems-First Integrals of the Motion	293	(2)
8.4 Equations of Motion	295	(4)
8.5 Orbits in a Central Field	299	(1)
8.6 Centrifugal Energy and the Effective Potential	300	(3)
8.7 Planetary Motion-Kepler's Problem	303	(6)
8.8 Orbital Dynamics	309	(8)
8.9 Apsidal Angles and Precession (optional)	317	(4)
8.10 Stability of Circular Orbits (optional)	321	(7)
Problems	328	(5)
9 DYNAMICS OF A SYSTEM OF PARTICLES	333	(48)
9.1 Introduction	333	(1)
9.2 Center of Mass	334	(2)
9.3 Linear Momentum of the System	336	(4)
9.4 Angular Momentum of the System	340	(4)
9.5 Energy of the System	344	(6)
9.6 Elastic Collisions of Two Particles	350	(7)
9.7 Kinematics of Elastic Collisions	357	(6)
9.8 Inelastic Collisions	363	(4)
9.9 Cross Sections	367	(5)
9.10 Rutherford Scattering Formula	372	(2)
Problems	374	(7)
10 MOTION IN A NONINERTIAL REFERENCE FRAME	381	(23)
10.1 Introduction	381	(1)
10.2 Rotating Coordinate Systems	382	(3)
10.3 Centrifugal and Coriolis Forces	385	(4)
10.4 Motion Relative to the Earth	389	(13)
Problems	402	(2)
11 DYNAMICS OF RIGID BODIES	404	(55)
11.1 Introduction	404	(1)
11.2 Inertia Tensor	405	(5)
11.3 Angular Momentum	410	(4)
11.4 Principal Axes of Inertia	414	(5)
11.5 Moments of Inertia for Different Body Coordinate Systems	419	(4)
11.6 Further Properties of the Inertia Tensor	423	(8)
11.7 Eulerian Angles	431	(4)
11.8 Euler's Equations for a Rigid Body	435	(5)
11.9 Force-Free Motion of a Symmetric Top	440	(5)
11.10 Motion of a Symmetric Top with One Point Fixed	445	(6)
11.11 Stability of Rigid-Body Rotations	451	(3)
Problems	454	(5)
12 COUPLED OSCILLATIONS	459	(44)
12.1 Introduction	459	(1)
12.2 Two Coupled Harmonic Oscillators	460	(4)
12.3 Weak Coupling	464	(2)
12.4 General Problem of Coupled Oscillations	466	(6)
12.5 Orthogonality of the Eigenvectors (optional)	472	(2)
12.6 Normal Coordinates	474	(8)
12.7 Molecular Vibrations	482	(4)
12.8 Three Linearly Coupled Plane Pendula-An Example of Degeneracy	486	(3)
12.9 The Loaded String	489	(10)
Problems	499	(4)
13 CONTINUOUS SYSTEMS; WAVES	503	(33)
13.1 Introduction	503	(1)
13.2 Continuous String as a Limiting Case of the Loaded String	504	(3)
13.3 Energy of a Vibrating String	507	(3)
13.4 Wave Equation	510	(2)
13.5 Forced and Damped Motion	512	(3)
13.6 General Solutions of the Wave Equation	515	(3)
13.7 Separation of the Wave Equation	518	(6)
13.8 Phase Velocity, Dispersion, and Attenuation	524	(5)
13.9 Group Velocity and Wave Packets	529	(4)
Problems	533	(3)
14 THE SPECIAL THEORY OF RELATIVITY	536	(43)
14.1 Introduction	536	(1)
14.2 Galilean Invariance	537	(2)
14.3 Lorentz Transformation	539	(6)
14.4 Experimental Verification of the Special Theory	545	(3)
14.5 Relativistic Doppler Effect	548	(3)
14.6 Twin Paradox	551	(2)
14.7 Relativistic Momentum	553	(3)
14.8 Energy	556	(4)
14.9 Spacetime and Four-Vectors	560	(8)
14.10 Lagrangian Function in Special Relativity	568	(2)
14.11 Relativistic Kinematics	570	(4)
Problems	574	(5)
APPENDICES
A TAYLOR'S THEOREM	579	(6)
Problems	583	(2)
B ELLIPTIC INTEGRALS	585	(5)
Problems	589	(1)
C ORDINARY DIFFERENTIAL EQUATIONS OF SECOND ORDER	590	(9)
C.1 Linear Homogeneous Equations	590	(4)
C.2 Linear Inhomogeneous Equations	594	(3)
Problems	597	(2)
D USEFUL FORMULAS	599	(5)
D.1 Binomial Expansion	599	(1)
D.2 Trigonometric Relations	600	(1)
D.3 Trigonometric Series	601	(1)
D.4 Exponential and Logarithmic Series	601	(1)
D.5 Complex Quantities	601	(1)
D.6 Hyperbolic Functions	602	(1)
Problems	603	(1)
E Useful Integrals	604	(4)
E.1 Algebraic Functions	604	(1)
E.2 Trigonometric Functions	605	(1)
E.3 Gamma Functions	606	(2)
F Differential Relations in Different Coordinate Systems	608	(4)
F.1 Rectangular Coordinates	608	(1)
F.2 Cylindrical Coordinates	608	(2)
F.3 Spherical Coordinates	610	(2)
G A "Proof" of the Relation XXX = XXX	612	(2)
H Numerical Solution for Example 2.7	614	(3)
Selected References	617	(2)
Bibliography	619	(4)
Answers to Even-Numbered Problems	623	(7)
Index	630