Bazanski / Zorski Foundations of Mechanics

1. Auflage 2013
ISBN: 978-1-4832-9161-1
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 28, 614 Seiten, Web PDF

Reihe: Studies in Applied Mechanics

In the last three decades the field of mechanics has seen spectacular progress due to the demand for applications in problems of cosmology, thermonuclear fusion, metallurgy, etc. This book provides a broad and thorough overview on the foundations of mechanics. It discusses theoretical mechanics and continuum mechanics, as well as phenomenological thermodynamics, quantum mechanics and relativistic mechanics. Each chapter presents the basic physical facts of interest without going into details and derivations and without using advanced mathematical formalism. The first part constitutes a classical exposition of Lagrange's and Hamilton's analytical mechanics on which most of the continuum theory is based. The section on continuum mechanics focuses mainly on the axiomatic foundations, with many pointers for further research in this area. Special attention is given to modern continuum thermodynamics, both for the foundations and applications. A section on quantum mechanics is also included, since the phenomenological description of various quantum phenomena is becoming of increasing importance. The work will prove indispensable to engineers wishing to keep abreast of recent theoretical advances in their field, as well as initiating and guiding future research.

Autoren/Hrsg.

Bazanski, Stanislaw

Zorski, H.

Inhaltsverzeichnis

1;Front Cover;1
2;Foundations of Mechanics;4
3;Copyright Page;5
4;Table of Contents;8
5;Preface;6
6;Part I: Analytical Mechanics (Roman Gutowski);16
6.1;Chapter 1. Constrained mechanical systems;18
6.1.1;1.1 Constrained and free mechanical systems;18
6.1.2;1.2 Holonomic, non-holonomic, scleronomous, and rheonomous constraints;19
6.1.3;1.3 Extending the concept of constraints. Constraints in controlled mechanical systems;24
6.2;Chapter 2. Variational principles of mechanics;28
6.2.1;2.1 Virtual displacements;28
6.2.2;2.2 Ideal constraints;30
6.2.3;2.3 D'Alembert's principle, Jourdain's principle, and Gauss's principle;32
6.2.4;2.4 The principle of virtual work and the foundations of analytical statics;41
6.2.5;2.5 Construction of equations of motion of mechanical systems based on differential variational principles;45
6.2.6;2.6 Generalized coordinates and velocities and quasi-coordinates and quasi-velocities;48
6.2.7;2.7 Hamilton's principle;57
6.2.8;2.8 Asynchronous variation. The Maupertuis-Lagrange principle;59
6.3;Chapter 3. Equations of motion of mechanical systems in Lagrange variables and quasi-coordinates;64
6.3.1;3.1 Lagrange equations of the second kind for holonomic systems;64
6.3.2;3.2 The energy of a system in generalized coordinates;69
6.3.3;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
6.3.4;3.4 Boltzmann-Hamel equations for holonomic systems in quasi-coordinates;86
6.3.5;3.5 Equations of motion for non-holonomic systems in generalized coordinates and quasi-coordinates;90
6.4;Chapter 4. Equations of motion of material systems in canonical variables;97
6.4.1;4.1 Canonical variables. Hamilton's function;97
6.4.2;4.2 Equations of motion of holonomic systems in canonical variables;100
6.4.3;4.3 Equations of motion of non-holonomic systems in canonical variables;104
6.4.4;4.4 The Hamilton-Jacobi method of investigating the motion of material systems, and its connection with the canonical equations of motion;106
6.5;Chapter 5. Canonical transformations;114
6.5.1;5.1 Finite and infinitely small canonical transformations;114
6.5.2;5.2 The connection between canonical transformations and the Hamilton-Jacobi theory;119
6.6;Chapter 6. Integral invariants and conservation laws;123
6.6.1;6.1 Systems of differential equations of motion which have integral invariants;123
6.6.2;6.2 The relationship between integral invariants and canonical transformations;127
6.6.3;6.3 Phase fluid and the hydrodynamic interpretation of integral invariants;128
6.6.4;6.4 Conservation laws in classical mechanics. The Noether theorem;131
6.7;Bibliography;134
7;Part II: Relativistic Mechanics (Stanislaw Leon Bazanski),;136
7.1;Chapter 1. Physical origin of the special theory of relativity;138
7.1.1;1.1 Development of early ideas about time and space;138
7.1.2;1.2 The Michelson-Morley experiment;140
7.1.3;1.3 Aberration;142
7.1.4;1.4 Fizeau's experiment;142
7.1.5;1.5 Precursors of new views on time and space;145
7.1.6;1.6 The approach proposed by Einstein;146
7.2;Chapter 2. Galilean space-time;148
7.2.1;2.1 Fundamental assumptions;148
7.2.2;2.2 The structure of space-time;149
7.2.3;2.3 Galilean transformations;151
7.2.4;2.4 The description of motion and the ether in Galilean space-time;152
7.3;Chapter 3. Basic space-time concepts of the special theory of relativity;154
7.3.1;3.1 Postulates;154
7.3.2;3.2 The physical construction of the basic space-time concepts;154
7.3.3;3.3 The composition of velocities;167
7.3.4;3.4 The Lorentz transformations;171
7.3.5;3.5 Length contraction;172
7.4;Chapter 4. Minkowski space-time;175
7.4.1;4.1 The structure of Minkowski space-time;175
7.4.2;4.2 Isometries of the Minkowski vector space;177
7.4.3;4.3 The Poincare transformations;185
7.4.4;4.4 Minkowski space as a model of space-time;187
7.4.5;4.5 The Minkowski diagram;188
7.4.6;4.6 Invariant space-time submanifolds;195
7.5;Chapter 5. Relativistic kinematics;197
7.5.1;5.1 The proper time of an arbitrary observer;197
7.5.2;5.2 The description of motion of a point particle;200
7.5.3;5.3 Description of motion in an instantaneous rest tetrad;207
7.6;Chapter 6. Dynamics of a material point;210
7.6.1;6.1 Postulates;210
7.6.2;6.2 Dynamics of a point particle;212
7.6.3;6.3 Examples;217
7.7;Chapter 7. Conservation principles;221
7.7.1;7.1 The Noether theorem for the dynamics of point particles;221
7.7.2;7.2 The Noether equation for relativistic dynamics;228
7.7.3;7.3 Dynamic symmetries corresponding to Poincare transformations;229
7.7.4;7.4 Angular momentum and the centre of mass of a system of free particles;233
7.7.5;7.5 An application of the second Noether theorem;238
7.8;Chapter 8. Equations of motion;240
7.8.1;8.1 The second principle of dynamics;240
7.8.2;8.2 Equations of motion in an instantaneous rest tetrad;242
7.8.3;8.3 The motion of a particle with internal angular momentum;244
7.9;Chapter 9. Canonical formalism;248
7.9.1;9.1 Two formulations of the problem;248
7.9.2;9.2 The fundamental lemma;249
7.9.3;9.3 Inhomogeneous formalism with coordinate time;250
7.9.4;9.4 Difficulties of the homogeneous formalism;252
7.9.5;9.5 Generalization of the fundamental lemma;255
7.9.6;9.6 Homogeneous formalism with proper time;257
7.9.7;9.7 The equivalence of both formalisms;259
7.9.8;9.8 Examples;260
7.9.9;9.9 The Hamilton-Jacobi equation;262
7.10;Chapter 10. Comments on the relativistic many-body problem;267
7.10.1;10.1 Relativistic mechanics and field theory;267
7.10.2;10.2 The no-interaction theorems;269
7.10.3;10.3 The dynamics of a system of many point particles;271
7.11;Bibliography;278
8;Part III: Quantum Mechanics (Jan Stawianowski),;280
8.1;Introduction;282
8.2;Chapter 1. Basic concepts of quantum mechanics. Historical origins;285
8.2.1;1.1 Principles of analytical mechanics;285
8.2.2;1.2 The Hamilton-Jacobi theory;297
8.2.3;1.3 Bohr-Sommerfeld conditions and the heuristics of quantization;306
8.3;Chapter 2. Quantum mechanics of a material point. Wave mechanics;316
8.3.1;2.1 Fundamental postulates of wave mechanics. Physical interpretation of the formalism;316
8.3.2;2.2 Quantization of material point mechanics. Position, momentum, and angular momentum;333
8.3.3;2.3 Dynamics, Schrödinger's equation, and the structure of the spectrum;350
8.4;Chapter 3. General formulation of quantum mechanics and examples;368
8.4.1;3.1 Hilbert space formalism;368
8.4.2;3.2 Description of spin. Wave mechanics of particles with spin;374
8.4.3;3.3 The many-body problem, identical particles;379
8.4.4;3.4 Quantization in curvilinear coordinates;382
8.5;Chapter 4. Simple applications of quantum mechanics;384
8.5.1;4.1 The meaning of exact solutions;384
8.5.2;4.2 The one-dimensional harmonic oscillator;384
8.5.3;4.3 The smooth potential well;387
8.5.4;4.4 The two-body problem, motion in a central field;389
8.5.5;4.5 The hydrogen-like atom;395
8.6;Chapter 5. Some approximate methods and their applications;399
8.6.1;5.1 Time-independent perturbation theory. The helium atom;399
8.6.2;5.2 Time-dependent perturbation theory. Interaction of atoms with an electromagnetic field;404
8.6.3;5.3 The Born-Oppenheimer method;412
8.6.4;5.4 The quasi-classical WKB method;414
8.7;Bibliography;418
9;Part IV: Mechanics of Continuous Media (Czeslaw Wozniak\;420
9.1;Introduction;422
9.2;Chapter 1. Basic concepts;424
9.2.1;1.1 Bodies and motions;424
9.2.2;1.2 Mass;428
9.2.3;1.3 Forces;428
9.2.4;1.4 Heat supply;429
9.2.5;1.5 Temperature, internal energy, and specific entropy;430
9.3;Chapter 2. Fundamental principles;432
9.3.1;2.1 Mass conservation principle;432
9.3.2;2.2 Principle of balance of momentum;432
9.3.3;2.3 Principle of balance of angular momentum;433
9.3.4;2.4 Principle of balance of energy;434
9.3.5;2.5 Dissipation principle;434
9.4;Chapter 3. Investigation of the balance principles;436
9.4.1;3.1 General balance principle;436
9.4.2;3.2 Flux field;437
9.4.3;3.3 Spatial forms of the balance principle;438
9.4.4;3.4 Referential forms of the balance principle;439
9.4.5;3.5 Conditions on singular surfaces;439
9.5;Chapter 4. General field equations;442
9.5.1;4.1 Continuity equation;442
9.5.2;4.2 Equations of motion: spatial description;443
9.5.3;4.3 Equations of motion: referential description;445
9.5.4;4.4 Energy equation;446
9.5.5;4.5 Local dissipation inequality;448
9.5.6;4.6 Strain and strain rate relations;449
9.6;Chapter 5. Materials;452
9.6.1;5.1 Basic ideas;452
9.6.2;5.2 The concept of history;455
9.6.3;5.3 Time-local properties and internal variables;456
9.6.4;5.4 Simple materials. Thermoelastic materials;458
9.6.5;5.5 Differential-type materials. Viscoelastic materials;459
9.6.6;5.6 Elastic/viscoplastic materials;460
9.6.7;5.7 Newtonian fluids;462
9.7;Chapter 6. Constraints and loadings;466
9.7.1;6.1 Basic ideas;466
9.7.2;6.2 Constraint responses;469
9.7.3;6.3 Some special constraints;473
9.7.4;6.4 Local boundary interactions;475
9.7.5;6.5 Constitutive internal constraints;476
9.8;Chapter 7. Specialized theories;481
9.8.1;7.1 Theory of finite thermoelastic deformations;481
9.8.2;7.2 Linearized theories in solid mechanics;484
9.8.3;7.3 Theory of Newtonian fluids;488
9.8.4;7.4 Structural mechanics theories;491
9.9;Final remarks;497
9.10;Bibliography;498
10;Part V: Phenomenological thermodynamics (Krzysztof Wilmanski),;500
10.1;Chapter 1. Introduction;502
10.2;Chapter 2. Fundamentals of abstract phenomenological thermodynamics;507
10.2.1;2.1 Preliminary discussion;507
10.2.2;2.2 Neoclassical thermodynamics of an isolated system;511
10.2.3;2.3 Neoclassical thermodynamics of thermodynamic subsystems;520
10.2.4;2.4 The scalar balance equation;524
10.2.5;2.5 Final remarks;530
10.3;Chapter 3. Thermodynamics of thermomechanical materials;532
10.3.1;3.1 The balance equations in the local theory of continuous media;532
10.3.2;3.2 The Clausius-Duhem inequality;540
10.3.3;3.3 The thermodynamics of a rigid heat conductor;546
10.3.4;3.4 The I-Shih Liu method—the Lagrange multipliers;552
10.3.5;3.5 The influence of body forces and radiation;564
10.3.6;3.6 General theory of materials;566
10.3.7;3.7 Examples;569
10.4;Chapter 4. Comments on the second law of thermodynamics;576
10.4.1;4.1 Introduction;576
10.4.2;4.2 The second law of thermodynamics in Caratheodory's formulation;577
10.4.3;4.3 The identities of classical thermostatics. Thermodynamic potentials;587
10.4.4;4.4 Cyclic processes and heat-engine efficiency;590
10.4.5;4.5 The second law of thermodynamics in Day's formulation;600
10.5;Bibliography;603
11;Index;605