Analytical Mechanics

by Louis N. Hand , Janet D. Finch

• ISBN13: 9780521573276
• ISBN10: 0521573270
• Format: Hardcover
• Copyright: 1998-11-13
• Publisher: Cambridge University Press

Summary

This introductory undergraduate text provides a detailed introduction to the key analytical techniques of classical mechanics, one of the cornerstones of physics. It deals with all the important subjects encountered in an undergraduate course and thoroughly prepares the reader for further study at graduate level. The authors set out the fundamentals of Lagrangian and Hamiltonian mechanics early in the book and go on to cover such topics as linear oscillators, planetary orbits, rigid-body motion, small vibrations, nonlinear dynamics, chaos, and special relativity. A special feature is the inclusion of many "e-mail questions," which are intended to facilitate dialogue between the student and instructor. It includes many worked examples, and there are 250 homework exercises to help students gain confidence and proficiency in problem-solving. It is an ideal textbook for undergraduate courses in classical mechanics, and provides a sound foundation for graduate study.

Table of Contents

Preface	xi
1 LAGRANGIAN MECHANICS	1	(43)
1.1 Example and Review of Newton's Mechanics: A Block Sliding on an Inclined Plane	1	(2)
1.2 Using Virtual Work to Solve the Same Problem	3	(4)
1.3 Solving for the Motion of a Heavy Bead Sliding on a Rotating Wire	7	(3)
1.4 Toward a General Formula: Degrees of Freedom and Types of Constraints	10	(4)
1.5 Generalized Velocities: How to "Cancel the Dots"	14	(1)
1.6 Virtual Displacements and Virtual Work - Generalized Forces	14	(2)
1.7 Kinetic Energy as a Function of the Generalized Coordinates and Velocities	16	(2)
1.8 Conservative Forces: Definition of the Lagrangian L	18	(2)
1.9 Reference Frames	20	(1)
1.10 Definition of the Hamiltonian	21	(1)
1.11 How to Get Rid of Ignorable Coordinates	22	(1)
1.12 Discussion and Conclusions -- What's Next after You Get the EOM?	23	(1)
1.13 An Example of a Solved Problem	24	(1)
Summary of Chapter 1	25	(1)
Problems	26	(10)
Appendix A. About Nonholonomic Constraints	36	(5)
Appendix B. More about Conservative Forces	41	(3)
2 VARIATIONAL CALCULUS AND ITS APPLICATION TO MECHANICS	44	(37)
2.1 History	44	(2)
2.2 The Euler Equation	46	(5)
2.3 Relevance to Mechanics	51	(2)
2.4 Systems with Several Degrees of Freedom	53	(1)
2.5 Why Use the Variational Approach in Mechanics?	54	(2)
2.6 Lagrange Multipliers	56	(1)
2.7 Solving Problems with Explicit Holonomic Constraints	57	(5)
2.8 Nonintegrable Nonholonomic Constraints -- A Method that Works	62	(3)
2.9 Postscript on the Euler Equation with More Than One Independent Variable	65	(1)
Summary of Chapter 2	65	(1)
Problems	66	(9)
Appendix. About Maupertuis and What Came to Be Called "Maupertuis' Principle"	75	(6)
3 LINEAR OSCILLATORS	81	(42)
3.1 Stable or Unstable Equilibrium?	82	(5)
3.2 Simple Harmonic Oscillator	87	(3)
3.3 Damped Simple Harmonic Oscillator (DSHO)	90	(4)
3.4 An Oscillator Driven by an External Force	94	(2)
3.5 Driving Force Is a Step Function	96	(3)
3.6 Finding the Green's Function for the SHO	99	(4)
3.7 Adding up the Delta Functions -- Solving the Arbitary Force	103	(2)
3.8 Driving an Oscillator in Resonance	105	(5)
3.9 Relative Phase of the DSHO Oscillator with Sinusoidal Drive	110	(3)
Summary of Chapter 3	113	(1)
Problems	114	(9)
4 ONE-DIMENSIONAL SYSTEMS: CENTRAL FORCES AND THE KEPLER PROBLEM	123	(47)
4.1 The Motion of a "Generic" One-Dimensional System	123	(2)
4.2 The Grandfather's Clock	125	(5)
4.3 The History of the Kepler Problem	130	(3)
4.4 Solving the Central Force Problem	133	(8)
4.5 The Special Case of Gravitational Attraction	141	(2)
4.6 Interpretation of Orbits	143	(8)
4.7 Repulsive 1/r(2) Forces	151	(5)
Summary of Chapter 4	156	(1)
Problems	156	(11)
Appendix. Tables of Astrophysical Data	167	(3)
5 NOETHER'S THEOREM AND HAMILTONIAN DYNAMICS	170	(37)
5.1 Discovering Angular Momentum Conservation from Rotational Invariance	170	(2)
5.2 Noether's Theorem	172	(3)
5.3 Hamiltonian Dynamics	175	(1)
5.4 The Legendre Transformation	175	(5)
5.5 Hamilton's Equations of Motion	180	(4)
5.6 Liouville's Theorem	184	(5)
5.7 Momentum Space	189	(1)
5.8 Hamiltonian Dynamics in Accelerated Systems	190	(5)
Summary of Chapter 5	195	(1)
Problems	196	(6)
Appendix A. A General Proof of Liouville's Theorem Using the Jacobian	202	(2)
Appendix B. Poincare Recurrence Theorem	204	(3)
6 THEORETICAL MECHANICS: FROM CANONICAL TRANSFORMATIONS TO ACTION-ANGLE VARIABLES	207	(45)
6.1 Canonical Transformations	208	(5)
6.2 Discovering Three New Forms of the Generating Function	213	(4)
6.3 Poisson Brackets	217	(1)
6.4 Hamilton-Jacobi Equation	218	(12)
6.5 Action-Angle Variables for 1-D Systems	230	(5)
6.6 Integrable Systems	235	(2)
6.7 Invariant Tori and Winding Numbers	237	(2)
Summary of Chapter 6	239	(1)
Problems	240	(8)
Appendix. What Does "Symplectic" Mean?	248	(4)
7 ROTATING COORDINATE SYSTEMS	252	(31)
7.1 What Is a Vector?	253	(1)
7.2 Review: Infinitesimal Rotations and Angular Velocity	254	(5)
7.3 Finite Three-Dimensional Rotations	259	(1)
7.4 Rotated Reference Frames	259	(4)
7.5 Rotating Reference Frames	263	(1)
7.6 The Instantaneous Angular Velocity Omega	264	(3)
7.7 Fictitious Forces	267	(1)
7.8 The Tower of Pisa Problem	267	(4)
7.9 Why Do Hurricane Winds Rotate?	271	(1)
7.10 Foucault Pendulum	272	(3)
Summary of Chapter 7	275	(1)
Problems	276	(7)
8 THE DYNAMICS OF RIGID BODIES	283	(60)
8.1 Kinetic Energy of a Rigid Body	284	(2)
8.2 The Moment of Inertia Tensor	286	(5)
8.3 Angular Momentum of a Rigid Body	291	(1)
8.4 The Euler Equations for Force-Free Rigid Body Motion	292	(1)
8.5 Motion of a Torque-Free Symmetric Top	293	(6)
8.6 Force-Free Precession of the Earth: The "Chandler Wobble"	299	(1)
8.7 Definition of Euler Angles	300	(4)
8.8 Finding the Angular Velocity	304	(1)
8.9 Motion of Torque-Free Asymmetric Tops: Poinsot Construction	305	(8)
8.10 The Heavy Symmetric Top	313	(4)
8.11 Precession of the Equinoxes	317	(6)
8.12 Mach's Principle	323	(2)
Summary of Chapter 8	325	(1)
Problems	326	(7)
Appendix A. What Is a Tensor?	333	(3)
Appendix B. Symmetric Matrices Can Always Be Diagonalized by "Rotating the Coordinates"	336	(3)
Appendix C. Understanding the Earth's Equatorial Bulge	339	(4)
9 THE THEORY OF SMALL VIBRATIONS	343	(40)
9.1 Two Coupled Pendulums	344	(4)
9.2 Exact Lagrangian for the Double Pendulum	348	(4)
9.3 Single Frequency Solutions to Equations of Motion	352	(3)
9.4 Superimposing Different Modes; Complex Mode Amplitudes	355	(5)
9.5 Linear Triatomic Molecule	360	(3)
9.6 Why the Method Always Works	363	(4)
9.7 N Point Masses Connected by a String	367	(4)
Summary of Chapter 9	371	(2)
Problems	373	(7)
Appendix. What Is a Cofactor?	380	(3)
10 APPROXIMATE SOLUTIONS TO NONANALYTIC PROBLEMS	383	(40)
10.1 Stability of Mechanical Systems	384	(4)
10.2 Parametric Resonance	388	(10)
10.3 Lindstedt-Poincare Perturbation Theory	398	(3)
10.4 Driven Anharmonic Oscillator	401	(10)
Summary of Chapter 10	411	(2)
Problems	413	(10)
11 CHAOTIC DYNAMICS	423	(70)
11.1 Conservative Chaos -- The Double Pendulum: A Hamiltonian System with Two Degrees of Freedom	426	(2)
11.2 The Poincare Section	428	(5)
11.3 KAM Tori: The Importance of Winding Number	433	(3)
11.4 Irrational Winding Numbers	436	(3)
11.5 Poincare-Birkhoff Theorem	439	(3)
11.6 Linearizing Near a Fixed Point: The Tangent Map and the Stability Matrix	442	(4)
11.7 Following Unstable Manifolds: Homoclinic Tangles	446	(3)
11.8 Lyapunov Exponents	449	(2)
11.9 Global Chaos for the Double Pendulum	451	(1)
11.10 Effect of Dissipation	452	(1)
11.11 Damped Driven Pendulum	453	(10)
11.12 Fractals	463	(5)
11.13 Chaos in the Solar System	468	(6)
Student Projects	474	(7)
Appendix. The Logistic Map: Period-Doubling Route to Chaos; Renormalization	481	(12)
12 SPECIAL RELATIVITY	493	(66)
12.1 Space-Time Diagrams	495	(3)
12.2 The Lorentz Transformation	498	(3)
12.3 Simultaneity Is Relative	501	(2)
12.4 What Happens to y and z if We Move Parallel to the X Axis?	503	(1)
12.5 Velocity Transformation Rules	504	(1)
12.6 Observing Light Waves	505	(7)
12.7 What Is Mass?	512	(1)
12.8 Rest Mass Is a Form of Energy	513	(4)
12.9 How Does Momentum Transform?	517	(2)
12.10 More Theoretical "Evidence" for the Equivalence of Mass and Energy	519	(2)
12.11 Mathematics of Relativity: Invariants and Four-Vectors	521	(5)
12.12 A Second Look at the Energy-Momentum Four-Vector	526	(3)
12.13 Why Are There Both Upper and Lower Greek Indices?	529	(1)
12.14 Relativistic Lagrangian Mechanics	530	(3)
12.15 What Is the Lagrangian in an Electromagnetic Field?	533	(2)
12.16 Does a Constant Force Cause Constant Acceleration?	535	(2)
12.17 Derivation of the Lorentz Force from the Lagrangian	537	(2)
12.18 Relativistic Circular Motion	539	(1)
Summary of Chapter 12	540	(1)
Problems	541	(13)
Appendix. The Twin Paradox	554	(5)
Bibliography	559	(4)
References	563	(2)
Index	565