Foundations of Mechanics

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ISBN: 9780444987006 | 0444987002
Cover: Hardcover
Copyright: 11/1/1992

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Preface
Analytical Mechanics	p. 1
Constrained mechanical systems	p. 3
Constrained and free mechanical systems	p. 3
Holonomic, non-holonomic, scleronomous, and rheonomous constraints	p. 4
Extending the concept of constraints. Constraints in controlled mechanical systems	p. 9
Variational principles of mechanics	p. 13
Virtual displacements	p. 13
Ideal constraints	p. 15
D'Alembert's principle, Jourdain's principle, and Gauss's principle	p. 17
The principle of virtual work and the foundations of analytical statics	p. 26
Construction of equations of motion of mechanical systems based on differential variational principles	p. 30
Generalized coordinates and velocities and quasi-coordinates and quasi-velocities	p. 33
Hamilton's principle	p. 42
Asynchronous variation. The Maupertuis-Lagrange principle	p. 44
Equations of motion of mechanical systems in Lagrange variables and quasi-coordinates	p. 49
Lagrange equations of the second kind for holonomic systems	p. 49
The energy of a system in generalized coordinates	p. 54
Lagrange's equations of the second kind in the form of ordinary differential equations. Invariance of Lagrange's equations of the second kind	p. 66
Boltzmann-Hamel equations for holonomic systems in quasi-coordinates	p. 71
Equations of motion for non-holonomic systems in generalized coordinates and quasi-coordinates	p. 75
Equations of motion of material systems in canonical variables	p. 82
Canonical variables. Hamilton's function	p. 82
Equations of motion of holonomic systems in canonical variables	p. 85
Equations of motion of non-holonomic systems in canonical variables	p. 89
The Hamilton-Jacobi method of investigating the motion of material systems, and its connection with the canonical equations of motion	p. 91
Canonical transformations	p. 99
Finite and infinitely small canonical transformations	p. 99
The connection between canonical transformations and the Hamilton-Jacobi theory	p. 104
Integral invariants and conservation laws	p. 108
Systems of differential equations of motion which have integral invariants	p. 108
The relationship between integral invariants and canonical transformations	p. 112
Phase fluid and the hydrodynamic interpretation of integral invariants	p. 113
Conservation laws in classical mechanics. The Noether theorem	p. 116
Bibliography	p. 119
Relativistic Mechanics	p. 121
Physical origin of the special theory of relativity	p. 123
Development of early ideas about time and space	p. 123
The Michelson-Morley experiment	p. 125
Aberration	p. 127
Fizeau's experiment	p. 127
Precursors of new views on time and space	p. 130
The approach proposed by Einstein	p. 131
Galilean space-time	p. 133
Fundamental assumptions	p. 133
The structure of space-time	p. 134
Galilean transformations	p. 136
The description of motion and the ether in Galilean space-time	p. 137
Basic space-time concepts of the special theory of relativity	p. 139
Postulates	p. 139
The physical construction of the basic space-time concepts	p. 139
Local time of inertial observers	p. 140
The relativity of simultaneity and dilation of time	p. 148
The clock paradox	p. 151
The composition of velocities	p. 152
The Lorentz transformations	p. 156
Length contraction	p. 157
Minkowski space-time	p. 160
The structure of Minkowski space-time	p. 160
Isometries of the Minkowski vector space	p. 162
The Poincare transformations	p. 170
Minkowski space as a model of space-time	p. 172
The Minkowski diagram	p. 173
The interpretation of Lorentz boosts	p. 175
The interpretation of various kinematic facts	p. 178
Invariant space-time submanifolds	p. 180
Relativistic kinematics	p. 182
The proper time of an arbitrary observer	p. 182
The description of motion of a point particle	p. 185
Velocity	p. 186
Acceleration	p. 188
Description of motion in an instantaneous rest tetrad	p. 192
Dynamics of a material point	p. 195
Postulates	p. 195
Dynamics of a point particle	p. 197
A free particle	p. 197
An interacting particle	p. 198
Lagrange equations of relativistic dynamics	p. 201
Examples	p. 202
Conservation principles	p. 206
The Noether theorem for the dynamics of point particles	p. 206
The Noether equation for relativistic dynamics	p. 213
Dynamic symmetries corresponding to Poincare transformations	p. 214
Angular momentum and the centre of mass of a system of free particles	p. 218
An application of the second Noether theorem	p. 223
Equations of motion	p. 225
The second principle of dynamics	p. 225
Equations of motion in an instantaneous rest tetrad	p. 227
The motion of a particle with internal angular momentum	p. 229
Canonical formalism	p. 233
Two formulations of the problem	p. 233
The fundamental lemma	p. 234
Inhomogeneous formalism with coordinate time	p. 235
Difficulties of the homogeneous formalism	p. 237
Generalization of the fundamental lemma	p. 240
Homogeneous formalism with proper time	p. 242
The equivalence of both formalisms	p. 244
Examples	p. 245
The Hamilton-Jacobi equation	p. 247
Comments on the relativistic many-body problem	p. 252
Relativistic mechanics and field theory	p. 252
The no-interaction theorems	p. 254
The dynamics of a system of many point particles	p. 256
Bibliography	p. 263
Quantum Mechanics	p. 265
Introduction	p. 267
Basic concepts of quantum mechanics. Historical origins	p. 270
Principles of analytical mechanics	p. 270
The Hamilton-Jacobi theory	p. 282
Bohr-Sommerfeld conditions and the heuristics of quantization	p. 291
Quantum mechanics of a material point. Wave mechanics	p. 301
Fundamental postulates of wave mechanics. Physical interpretation of the formalism	p. 301
Quantization of material point mechanics. Position, momentum, and angular momentum	p. 318
Dynamics, Schrodinger's equation, and the structure of the spectrum	p. 335
General formulation of quantum mechanics and examples	p. 353
Hilbert space formalism	p. 353
Description of spin. Wave mechanics of particles with spin	p. 359
The many-body problem, identical particles	p. 364
Quantization in curvilinear coordinates	p. 367
Simple applications of quantum mechanics	p. 369
The meaning of exact solutions	p. 369
The one-dimensional harmonic oscillator	p. 369
The smooth potential well	p. 372
The two-body problem, motion in a central field	p. 374
The hydrogen-like atom	p. 380
Some approximate methods and their applications	p. 384
Time-independent perturbation theory. The helium atom	p. 384
Time-dependent perturbation theory. Interaction of atoms with an electromagnetic field	p. 389
The Born-Oppenheimer method	p. 397
The quasi-classical WKB method	p. 399
Bibliography	p. 403
Mechanics of Continuous Media	p. 405
Introduction	p. 407
Basic concepts	p. 409
Bodies and motions	p. 409
Mass	p. 413
Forces	p. 413
Heat supply	p. 414
Temperature, internal energy, and specific entropy	p. 415
Fundamental principles	p. 417
Mass conservation principle	p. 417
Principle of balance of momentum	p. 417
Principle of balance of angular momentum	p. 418
Principle of balance of energy	p. 419
Dissipation principle	p. 419
Investigation of the balance principles	p. 421
General balance principle	p. 421
Flux field	p. 422
Spatial forms of the balance principle	p. 423
Referential forms of the balance principle	p. 424
Conditions on singular surfaces	p. 424
General field equations	p. 427
Continuity equation	p. 427
Equations of motion: spatial description	p. 428
Equations of motion: referential description	p. 430
Energy equation	p. 431
Local dissipation inequality	p. 433
Strain and strain rate relations	p. 434
Materials	p. 437
Basic ideas	p. 437
The concept of history	p. 440
Time-local properties and internal variables	p. 441
Simple materials. Thermoelastic materials	p. 443
Differential-type materials. Viscoelastic materials	p. 444
Elastic/viscoplastic materials	p. 445
Newtonian fluids	p. 447
Constraints and loadings	p. 451
Basic ideas	p. 451
Constraint responses	p. 454
Some special constraints	p. 458
Local boundary interactions	p. 460
Constitutive internal constraints	p. 461
Specialized theories	p. 466
Theory of finite thermoelastic deformations	p. 466
Linearized theories in solid mechanics	p. 469
Theory of Newtonian fluids	p. 473
Structural mechanics theories	p. 476
Final remarks	p. 482
Bibliography	p. 483
Phenomenological thermodynamics	p. 485
Introduction	p. 487
Fundamentals of abstract phenomenological thermodynamics	p. 492
Preliminary discussion	p. 492
Neoclassical thermodynamics of an isolated system	p. 496
Neoclassical thermodynamics of thermodynamic subsystems	p. 505
The scalar balance equation	p. 509
Final remarks	p. 515
Thermodynamics of thermomechanical materials	p. 517
The balance equations in the local theory of continuous media	p. 517
The Clausius-Duhem inequality	p. 525
The thermodynamics of a rigid heat conductor	p. 531
The I-Shih Liu method--the Lagrange multipliers	p. 537
The influence of body forces and radiation	p. 549
General theory of materials	p. 551
Examples	p. 554
The continuity of enthalpy in the classical theory of phase transitions	p. 555
The constitutive relation for a real gas and the phase transition in it	p. 557
Comments on the second law of thermodynamics	p. 561
Introduction	p. 561
The second law of thermodynamics in Caratheodory's formulation	p. 562
The identities of classical thermostatics. Thermodynamic potentials	p. 572
Cyclic processes and heat-engine efficiency	p. 575
The second law of thermodynamics in Day's formulation	p. 585
Bibliography	p. 588
Index	p. 590
Table of Contents provided by Blackwell. All Rights Reserved.

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