Foundations of Mechanics (eBook)

Front Cover 1
Foundations of Mechanics 4
Copyright Page 5
Table of Contents 8
Preface 6
Part I: Analytical Mechanics (Roman Gutowski) 16
Chapter 1. Constrained mechanical systems 18
1.1 Constrained and free mechanical systems 18
1.2 Holonomic, non-holonomic, scleronomous, and rheonomous constraints 19
1.3 Extending the concept of constraints. Constraints in controlled mechanical systems 24
Chapter 2. Variational principles of mechanics 28
2.1 Virtual displacements 28
2.2 Ideal constraints 30
2.3 D'Alembert's principle, Jourdain's principle, and Gauss's principle 32
2.4 The principle of virtual work and the foundations of analytical statics 41
2.5 Construction of equations of motion of mechanical systems based on differential variational principles 45
2.6 Generalized coordinates and velocities and quasi-coordinates and quasi-velocities 48
2.7 Hamilton's principle 57
2.8 Asynchronous variation. The Maupertuis-Lagrange principle 59
Chapter 3. Equations of motion of mechanical systems in Lagrange variables and quasi-coordinates 64
3.1 Lagrange equations of the second kind for holonomic systems 64
3.2 The energy of a system in generalized coordinates 69
3.3 Lagrange's equations of the second kind in the form of ordinary differential equations. Invariance of Lagrange's equations of the second kind 81
3.4 Boltzmann-Hamel equations for holonomic systems in quasi-coordinates 86
3.5 Equations of motion for non-holonomic systems in generalized coordinates and quasi-coordinates 90
Chapter 4. Equations of motion of material systems in canonical variables 97
4.1 Canonical variables. Hamilton's function 97
4.2 Equations of motion of holonomic systems in canonical variables 100
4.3 Equations of motion of non-holonomic systems in canonical variables 104
4.4 The Hamilton-Jacobi method of investigating the motion of material systems, and its connection with the canonical equations of motion 106
Chapter 5. Canonical transformations 114
5.1 Finite and infinitely small canonical transformations 114
5.2 The connection between canonical transformations and the Hamilton-Jacobi theory 119
Chapter 6. Integral invariants and conservation laws 123
6.1 Systems of differential equations of motion which have integral invariants 123
6.2 The relationship between integral invariants and canonical transformations 127
6.3 Phase fluid and the hydrodynamic interpretation of integral invariants 128
6.4 Conservation laws in classical mechanics. The Noether theorem 131
Bibliography 134
Part II: Relativistic Mechanics (Stanislaw Leon Bazanski), 136
Chapter 1. Physical origin of the special theory of relativity 138
1.1 Development of early ideas about time and space 138
1.2 The Michelson-Morley experiment 140
1.3 Aberration 142
1.4 Fizeau's experiment 142
1.5 Precursors of new views on time and space 145
1.6 The approach proposed by Einstein 146
Chapter 2. Galilean space-time 148
2.1 Fundamental assumptions 148
2.2 The structure of space-time 149
2.3 Galilean transformations 151
2.4 The description of motion and the ether in Galilean space-time 152
Chapter 3. Basic space-time concepts of the special theory of relativity 154
3.1 Postulates 154
3.2 The physical construction of the basic space-time concepts 154
3.3 The composition of velocities 167
3.4 The Lorentz transformations 171
3.5 Length contraction 172
Chapter 4. Minkowski space-time 175
4.1 The structure of Minkowski space-time 175
4.2 Isometries of the Minkowski vector space 177
4.3 The Poincare transformations 185
4.4 Minkowski space as a model of space-time 187
4.5 The Minkowski diagram 188
4.6 Invariant space-time submanifolds 195
Chapter 5. Relativistic kinematics 197
5.1 The proper time of an arbitrary observer 197
5.2 The description of motion of a point particle 200
5.3 Description of motion in an instantaneous rest tetrad 207
Chapter 6. Dynamics of a material point 210
6.1 Postulates 210
6.2 Dynamics of a point particle 212
6.3 Examples 217
Chapter 7. Conservation principles 221
7.1 The Noether theorem for the dynamics of point particles 221
7.2 The Noether equation for relativistic dynamics 228
7.3 Dynamic symmetries corresponding to Poincare transformations 229
7.4 Angular momentum and the centre of mass of a system of free particles 233
7.5 An application of the second Noether theorem 238
Chapter 8. Equations of motion 240
8.1 The second principle of dynamics 240
8.2 Equations of motion in an instantaneous rest tetrad 242
8.3 The motion of a particle with internal angular momentum 244
Chapter 9. Canonical formalism 248
9.1 Two formulations of the problem 248
9.2 The fundamental lemma 249
9.3 Inhomogeneous formalism with coordinate time 250
9.4 Difficulties of the homogeneous formalism 252
9.5 Generalization of the fundamental lemma 255
9.6 Homogeneous formalism with proper time 257
9.7 The equivalence of both formalisms 259
9.8 Examples 260
9.9 The Hamilton-Jacobi equation 262
Chapter 10. Comments on the relativistic many-body problem 267
10.1 Relativistic mechanics and field theory 267
10.2 The no-interaction theorems 269
10.3 The dynamics of a system of many point particles 271
Bibliography 278
Part III: Quantum Mechanics (Jan Stawianowski), 280
Introduction 282
Chapter 1. Basic concepts of quantum mechanics. Historical origins 285
1.1 Principles of analytical mechanics 285
1.2 The Hamilton-Jacobi theory 297
1.3 Bohr-Sommerfeld conditions and the heuristics of quantization 306
Chapter 2. Quantum mechanics of a material point. Wave mechanics 316
2.1 Fundamental postulates of wave mechanics. Physical interpretation of the formalism 316
2.2 Quantization of material point mechanics. Position, momentum, and angular momentum 333
2.3 Dynamics, Schrödinger's equation, and the structure of the spectrum 350
Chapter 3. General formulation of quantum mechanics and examples 368
3.1 Hilbert space formalism 368
3.2 Description of spin. Wave mechanics of particles with spin 374
3.3 The many-body problem, identical particles 379
3.4 Quantization in curvilinear coordinates 382
Chapter 4. Simple applications of quantum mechanics 384
4.1 The meaning of exact solutions 384
4.2 The one-dimensional harmonic oscillator 384
4.3 The smooth potential well 387
4.4 The two-body problem, motion in a central field 389
4.5 The hydrogen-like atom 395
Chapter 5. Some approximate methods and their applications 399
5.1 Time-independent perturbation theory. The helium atom 399
5.2 Time-dependent perturbation theory. Interaction of atoms with an electromagnetic field 404
5.3 The Born-Oppenheimer method 412
5.4 The quasi-classical WKB method 414
Bibliography 418
Part IV: Mechanics of Continuous Media (Czeslaw Wozniak/ 420
Introduction 422
Chapter 1. Basic concepts 424
1.1 Bodies and motions 424
1.2 Mass 428
1.3 Forces 428
1.4 Heat supply 429
1.5 Temperature, internal energy, and specific entropy 430
Chapter 2. Fundamental principles 432
2.1 Mass conservation principle 432
2.2 Principle of balance of momentum 432
2.3 Principle of balance of angular momentum 433
2.4 Principle of balance of energy 434
2.5 Dissipation principle 434
Chapter 3. Investigation of the balance principles 436
3.1 General balance principle 436
3.2 Flux field 437
3.3 Spatial forms of the balance principle 438
3.4 Referential forms of the balance principle 439
3.5 Conditions on singular surfaces 439
Chapter 4. General field equations 442
4.1 Continuity equation 442
4.2 Equations of motion: spatial description 443
4.3 Equations of motion: referential description 445
4.4 Energy equation 446
4.5 Local dissipation inequality 448
4.6 Strain and strain rate relations 449
Chapter 5. Materials 452
5.1 Basic ideas 452
5.2 The concept of history 455
5.3 Time-local properties and internal variables 456
5.4 Simple materials. Thermoelastic materials 458
5.5 Differential-type materials. Viscoelastic materials 459
5.6 Elastic/viscoplastic materials 460
5.7 Newtonian fluids 462
Chapter 6. Constraints and loadings 466
6.1 Basic ideas 466
6.2 Constraint responses 469
6.3 Some special constraints 473
6.4 Local boundary interactions 475
6.5 Constitutive internal constraints 476
Chapter 7. Specialized theories 481
7.1 Theory of finite thermoelastic deformations 481
7.2 Linearized theories in solid mechanics 484
7.3 Theory of Newtonian fluids 488
7.4 Structural mechanics theories 491
Final remarks 497
Bibliography 498
Part V: Phenomenological thermodynamics (Krzysztof Wilmanski), 500
Chapter 1. Introduction 502
Chapter 2. Fundamentals of abstract phenomenological thermodynamics 507
2.1 Preliminary discussion 507
2.2 Neoclassical thermodynamics of an isolated system 511
2.3 Neoclassical thermodynamics of thermodynamic subsystems 520
2.4 The scalar balance equation 524
2.5 Final remarks 530
Chapter 3. Thermodynamics of thermomechanical materials 532
3.1 The balance equations in the local theory of continuous media 532
3.2 The Clausius-Duhem inequality 540
3.3 The thermodynamics of a rigid heat conductor 546
3.4 The I-Shih Liu method—the Lagrange multipliers 552
3.5 The influence of body forces and radiation 564
3.6 General theory of materials 566
3.7 Examples 569
Chapter 4. Comments on the second law of thermodynamics 576
4.1 Introduction 576
4.2 The second law of thermodynamics in Caratheodory's formulation 577
4.3 The identities of classical thermostatics. Thermodynamic potentials 587
4.4 Cyclic processes and heat-engine efficiency 590
4.5 The second law of thermodynamics in Day's formulation 600
Bibliography 603
Index 605