Theoretical Mechanics of Particles and Continua,9780486432618

Theoretical Mechanics of Particles and Continua

by ;
ISBN13: 9780486432618
ISBN10: 0486432610
Format: Paperback
Pub. Date: 2003-12-16
Publisher(s): Dover Publications
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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

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