Theoretical Mechanics of Particles and Continua

by Fetter, Alexander L.; Walecka, John Dirk

• ISBN13: 9780486432618
• ISBN10: 0486432610
• Format: Paperback
• Copyright: 2003-12-16
• Publisher: Dover Publications

Summary

This two-part text supplies a lucid, self-contained account of classical mechanics and provides a natural framework for introducing advanced mathematical concepts in physics. Topics include Lagrangian dynamics, Hamiltonian dynamics, fluids and sound and surface waves, more. 165 figures. 2 tables. 1980 edition.

Table of Contents

Preface	xv
Significant Names in Mechanics and Mathematical Physics	xvii
Basic Principles	1	(30)
Newton's Laws	1	(4)
Statement of Newton's Laws	1	(2)
Conservation Laws	3	(2)
Systems of Particles	5	(5)
Center-of-Mass Motion	5	(1)
Angular Momentum	6	(2)
Energy	8	(2)
Central Forces	10	(6)
Conservation Laws	10	(1)
Effective Potential	11	(2)
Inverse-Square Force: Kepler's Laws	13	(3)
Two-Body Motion with a Central Potential	16	(2)
Scattering	18	(13)
Hyperbolic Orbits in Gravitational Potential	19	(3)
General Scattering Orbits	22	(1)
Cross Section	23	(2)
Rutherford Scattering	25	(2)
Scattering by a Hard Sphere	27	(4)
Accelerated Coordinate Systems	31	(18)
Rotating Coordinate Systems	31	(2)
Infinitesimal Rotations	33	(3)
Accelerations	36	(1)
Translations and Rotations	37	(1)
Newton's Laws in Accelerated Coordinate Systems	38	(1)
Motion on the Surface of the Earth	39	(5)
Particle on a Scale	40	(1)
Falling Particle	41	(2)
Horizontal Motion	43	(1)
Foucault Pendulum	44	(5)
Lagrangian Dynamics	49	(37)
Constrained Motion and Generalized Coordinates	49	(3)
Constraints	49	(1)
Generalized Coordinates	50	(1)
Virtual Displacements	51	(1)
D'Alembert's Principle	52	(1)
Lagrange's Equations	53	(5)
Examples	58	(2)
Pendulum	58	(1)
Bead on a Rotating Wire Hoop	59	(1)
Calculus of Variations	60	(6)
Hamilton's Principle	66	(2)
Forces of Constraint	68	(10)
Pendulum	71	(2)
Atwood's Machine	73	(1)
One Cylinder Rolling on Another	74	(4)
Generalized Momenta and the Hamiltonian	78	(8)
Symmetry Principles and Conserved Quantities	78	(1)
The Hamiltonian	79	(7)
Small Oscillations	86	(48)
Formulation	86	(3)
Normal Modes	89	(12)
Simplest Case	89	(2)
Coupled Problem: Formulation	91	(1)
Linear Equations: A Review	92	(1)
Coupled Problem: Eigenvectors and Eigenvalues	93	(2)
Coupled Problem: General Solution	95	(1)
Matrix Notation	96	(2)
Modal Matrix	98	(1)
Normal Coordinates	99	(2)
Example: Coupled Pendulums	101	(7)
Example: Many Degrees of Freedom	108	(11)
Two N-Body Problems	108	(2)
Normal Modes	110	(9)
Transition from Discrete to Continuous Systems	119	(15)
Passage to the Continuum Limit	120	(1)
Direct Treatment of a Continuous String	120	(2)
General Solution to the Wave Equation with Specified Initial Conditions	122	(3)
Lagrangian for a Continuous String	125	(1)
Normal Coordinates	126	(2)
Hamilton's Principle for Continuous Systems	128	(6)
Rigid Bodies	134	(39)
General Theory	134	(9)
Motion with One Arbitrary Fixed Point	134	(3)
General Motion with No Fixed Point	137	(2)
Inertia Tensor	139	(1)
Principal Axes	140	(3)
Euler's Equations	143	(1)
Applications	144	(10)
Compound Pendulum: Kater's Pendulum and the Center of Percussion	144	(5)
Rolling and Sliding Billiard Ball	149	(2)
Torque-free Motion: Symmetric Top	151	(2)
Torque-free Motion: Asymmetric Top	153	(1)
Euler Angles	154	(2)
Symmetric Top: Torque-free Motion	156	(5)
Equations of Motion and First Integrals	157	(1)
Description of Motion in Inertial Frame	158	(3)
Symmetric Top: One Fixed Point in a Gravitational Field	161	(12)
Dynamical Equations	161	(2)
Effective Potential	163	(2)
Small Oscillations about Steady Motion	165	(8)
Hamiltonian Dynamics	173	(34)
Hamilton's Equations	173	(6)
Review of Lagrangian Dynamics	173	(2)
Hamiltonian Dynamics	175	(2)
Derivation of Hamilton's Equations from a Modified Hamilton's Principle	177	(2)
Example: Charged Particle in an Electromagnetic Field	179	(2)
Canonical Transformations	181	(3)
Hamilton-Jacobi Theory	184	(7)
Action-Angle Variables	191	(6)
Poisson Brackets	197	(10)
Basic Formulation	197	(1)
Transition to Quantum Mechanics	198	(9)
Strings	207	(64)
Review of Field Theory	207	(4)
D'Alembert's Solution to the Wave Equation	211	(8)
Solution for an Infinite String	211	(3)
Solution for a Finite String	214	(1)
Equivalence of d'Alembert's and Bernoulli's Solution	215	(4)
Eigenfunction Expansions	219	(7)
Variational Principle	226	(10)
Basic Formulation	227	(2)
Minimum Character of the Functional	229	(3)
Completeness of Eigenfunctions	232	(4)
Estimates of Lowest Eigenvalues; The Rayleigh-Ritz Approximation Method	236	(9)
General Theory	237	(2)
Example: Mass Point on a String	239	(6)
Green's Function in One Dimension	245	(6)
Eigenfunction Expansion	245	(2)
Construction from Solutions to Homogeneous Equations	247	(2)
Example: Uniform String with Fixed Endpoints	249	(2)
Perturbation Theory	251	(7)
General Theory	252	(2)
Expansion for Small Coupling Strength	254	(1)
Example: Mass Point on a String Revisited	255	(3)
Energy Flux	258	(13)
Continuity Equation for the Hamiltonian Density	258	(4)
Example: One-dimensional String	262	(1)
Transmission and Reflection at a Discontinuity in Density	263	(8)
Membranes	271	(19)
General Formulation	271	(3)
Specific Geometries	274	(16)
Rectangular Membrane	274	(5)
Circular Membrane	279	(4)
Variational Estimate of Lowest Drumhead Mode	283	(1)
Perturbation Theory for Nearly Circular Boundary	284	(6)
Sound Waves in Fluids	290	(67)
General Equations of Hydrodynamics	290	(15)
Formulation of Newton's Second Law	291	(3)
Conservation of Matter: The Continuity Equation	294	(2)
Conservation of Momentum: Stress Tensor and Euler's Equation	296	(2)
Conservation of Energy	298	(2)
Bernoulli's Theorem	300	(2)
Thomson's (Lord Kelvin's) Theorem on Circulation	302	(1)
Lagrangian for Isentropic Irrotational Flow	303	(2)
Sound Waves	305	(6)
Fundamental Equations	305	(3)
Standing Waves in Cavities	308	(3)
Fourier Transforms and Green's Functions in Three Dimensions	311	(9)
Screened Poisson Equation	312	(2)
Helmholtz Equation: Causality and Analyticity	314	(3)
Boundaries and the Method of Images	317	(3)
Radiation, Diffraction, and Scattering	320	(19)
Radiation from a Piston in a Wall	320	(5)
Diffraction in Kirchhoff's Approximation	325	(7)
Radiation from an Oscillating Sphere	332	(4)
Scattering by a Rigid Cylinder	336	(3)
Nonlinear Phenomena and Shock Waves	339	(18)
Traveling Waves	340	(3)
Example: Ideal Gas	343	(4)
Shock Waves	347	(10)
Surface Waves on Fluids	357	(49)
Tidal Waves	357	(9)
Equations of Motion	357	(2)
One-dimensional Waves	359	(4)
Two-dimensional Waves	363	(3)
Surface Waves	366	(17)
Formulation for Arbitrary Depths	367	(3)
Dispersion Relation	370	(4)
Energy	374	(2)
Group Velocity	376	(3)
Inclusion of Surface Tension	379	(4)
Initial-Value Problem	383	(10)
Surface Waves on Deep Water	384	(3)
Method of Stationary Phase	387	(2)
Application to Surface Waves	389	(4)
Solitary Waves	393	(13)
Extended Equation for Tidal Waves	395	(3)
Effective Nonlinear Wave Equation	398	(1)
Solitary Waves	399	(7)
Heat Conduction	406	(28)
Basic Equations	406	(4)
Examples	410	(7)
Separation of Variables	411	(2)
Thermal Waves in a Half Space	413	(2)
Infinite Domain: Fourier Transform	415	(2)
Laplace Transform	417	(17)
Inversion Theorem	417	(2)
Example: Half Space at Fixed Surface Temperature	419	(5)
Example: Sphere Heated Internally	424	(3)
Approximation Methods for Long and Short Times	427	(7)
Viscous Fluids	434	(25)
Viscous Stress Tensor	434	(11)
Basic Formulation	435	(3)
Navier-Stokes Equation	438	(3)
Energy Balance	441	(4)
Examples of Incompressible Flow	445	(6)
Steady Flow in a Channel or Pipe	445	(3)
Tangential Flow in a Half Space	448	(3)
Sound Waves in Viscous Fluids	451	(8)
Elastic Continua	459	(22)
Basic Formulation	459	(11)
Small Deformations	460	(4)
Stress Tensor	464	(4)
Elastic Energy	468	(2)
Dynamical Behavior	470	(11)
Equation of Motion	471	(4)
Elastic Waves in an Unbounded Medium	475	(6)
Appendix A Theory of Functions	481	(31)
A1 Complex Variables	481	(1)
A2 Functions of a Complex Variable	482	(4)
A3 Complex Integration	486	(2)
A4 Cauchy's Theorem	488	(5)
A5 Cauchy's Integral	493	(2)
Morera's Theorem	494	(1)
A6 Uniformly Convergent Series	495	(2)
Power Series	496	(1)
A7 Taylor's Theorem	497	(1)
A8 Laurent Series	498	(2)
A9 Theory of Residues	500	(4)
A10 Zeros of an Analytic Function	504	(2)
A11 Analytic Continuation	506	(6)
Appendix B Curvilinear Orthogonal Coordinates	512	(9)
Gradient	514	(1)
Divergence	514	(2)
Laplacian	516	(1)
Spherical Coordinates	516	(1)
Cylindrical Coordinates	516	(1)
Polar Coordinates in Two Dimensions	517	(4)
Appendix C Separation of Variables	521	(5)
Normal Modes in Polar Coordinates (Two Dimensions)	521	(1)
Normal Modes in Spherical Coordinates (Three Dimensions)	522	(2)
Normal Modes in Cylindrical Coordinates (Three Dimensions)	524	(2)
Appendix D Integral Representations and Special Functions	526	(22)
D1 The Γ Function	526	(5)
D2 Legendre Functions	531	(10)
α = l (an integer)	532	(3)
Arbitrary α	535	(1)
Legendre Functions of the Second Kind	535	(3)
Arbitrary α	538	(3)
D3 Bassel Functions	541	(7)
Appendix E Selected Mathematical Formulas	548	(6)
E1 The Γ Function	548	(1)
E2 Error Function	549	(1)
E3 Legendre Functions	549	(1)
Recursion and General Relations [also for Qα(z)]	549	(1)
Addition Formulas	549	(1)
Explicit Forms for l = 0, 1, 2, 3, . . . , ∞ and integral m	550	(1)
E4 Cylindrical Bessel Functions	550	(2)
Recursion Relations [also for Nv(z)]	550	(1)
Series and Approximate Forms (m is a non-negative integer)	551	(1)
E5 Spherical Bessel Functions	552	(2)
Specific Forms	552	(1)
Recursion Relations [also for nl(z)]	553	(1)
Appendix F Physical Constants	554	(2)
Appendix G Basic Texts and Monographs	556	(1)
Index	557