Mechanics, Tensors & Virtual Works

PREFACE......Page 6
CONTENTS......Page 7
1.1 POINT SPACE (OR AFFINE SPACE)......Page 15
1.2 FRAME OF REFERENCE AND BASIS......Page 16
2.1 DYNAM DEFINITION AND REDUCTION ELEMENTS......Page 17
2.2.1 Equality of Dynams......Page 20
2.2.2 Operations on Dynams......Page 21
2.2.3 Equiprojective Fields of Moments......Page 24
2.2.4 Invariants......Page 25
2.2.5 Reduction of a Vector System and Dynam......Page 26
2.3.1 Velocity Field......Page 31
2.3.2 Dynam of Velocities......Page 34
2.4 ACCELERATION VECTORS......Page 35
2.5 SLIDING VELOCITY......Page 40
3. EXERCISES......Page 41
CHAPTER 1. STATICS......Page 47
1.1.3 Moment of a Force......Page 48
1.1.4 Dynam of Mechanical Action......Page 49
1.2.1 External Forces......Page 50
1.2.2 Internal Forces......Page 51
1.3.1 Definitions and Conditions......Page 52
1.3.2 Particular Collections of Forces Applied to a Rigid Body......Page 56
1.4 TYPES OF EQUILIBRIUM......Page 58
1.5.1 Stress......Page 61
1.5.2 Contact Dynam......Page 62
1.5.3 Dry Friction and Coulomb’s Laws......Page 64
1.6 TYPES OF CONSTRAINTS......Page 68
1.6.1 Punctual Constraint......Page 69
1.6.2 Rectilinear Constraint......Page 70
1.6.3 Annular-Linear Constraint......Page 72
1.6.4 Ball-and-Socket Joint......Page 73
1.6.5 Plane Support......Page 74
1.6.6 Sliding Pivot......Page 75
1.6.7 Sliding Guide......Page 77
1.6.8 Screw Joint......Page 78
1.6.9 Pivot......Page 79
1.7 FREE-BODY DIAGRAM......Page 82
2. METHOD OF VIRTUAL WORK......Page 84
2.1.1 Number of Degrees of Freedom......Page 85
2.1.2 Generalized Coordinates......Page 86
2.1.3 Types of Constraints......Page 89
2.2.1 Generalized Coordinates......Page 92
2.2.2 Definition and Expressions of Virtual Displacements......Page 93
2.2.3 Virtual Velocity and Examples......Page 95
2.2.4 Virtual Fields and Dynams......Page 105
2.3.1 Definitions, Rigid Body Motion and Ideal Constraint......Page 108
2.3.2 Principle of Virtual Work (First Expression)......Page 112
2.3.3 Principle of Virtual Work (Second Expression)......Page 120
3. EXERCISES......Page 128
1.1.1 Linear Mapping......Page 148
1.2.1 Dual Space......Page 149
1.2.2 Expression of a Covector......Page 150
1.2.3 Einstein Summation Convention......Page 151
1.2.4 Change of Basis and Cobasis......Page 153
1.3.1 Tensor Product of Multilinear Forms......Page 156
1.3.2 Tensor of Type......Page 157
1.3.3 Tensor of Type......Page 158
1.3.4 Tensor of Type......Page 159
1.3.5 Tensor of Type......Page 162
1.3.6 Tensor of Type......Page 163
1.3.7 Tensor of Type......Page 164
2.1.1 Addition of Tensors......Page 169
2.1.3 Tensor Multiplication......Page 170
2.2.1 Contraction......Page 171
2.2.2 Tensor Criterion......Page 175
3.1.2 Fundamental Tensor......Page 177
3.2 CANONICAL ISOMORPHISM AND CONJUGATE TENSOR......Page 178
3.2.1 Canonical Isomorphism......Page 179
3.2.2 Conjugate Tensor and Reciprocal Basis......Page 180
3.2.3 Covariant and Contravariant Representations of Vectors......Page 183
3.2.4 Representations of Tensors of Order 2 and Contracted Products......Page 185
3.3 EUCLIDEAN VECTOR SPACES......Page 187
4. EXTERIOR ALGEBRA......Page 190
4.1.1 Definition of a p-Form......Page 191
4.1.2 Exterior Product of 1-Forms......Page 192
4.1.3 Expression of a p-Form......Page 193
4.1.4 Exterior Product of p-Forms......Page 197
4.1.5 Exterior Algebra......Page 198
4.2 q-VECTORS......Page 201
5. POINT SPACES......Page 204
5.1.2 Coordinate System and Frame of Reference......Page 205
5.1.3 Natural Frame......Page 207
5.2.1 Transformations of Curvilinear Coordinates......Page 210
5.2.2 Tensor Fields......Page 213
5.2.3 Metric Element......Page 214
5.3.1 Definition of Christoffel Symbols......Page 216
5.3.2 Ricci Identities and Christoffel Formulae......Page 219
5.4.1 Absolute Differential of a Vector, Covariant Derivatives......Page 220
5.4.2 Absolute Differential of a Tensor, Covariant Derivatives......Page 222
5.4.3 Geodesic and Euler’s Equations......Page 224
5.4.4 Parallel Transport......Page 225
5.4.5 Absolute Derivative of a Vector (Along a Curve)......Page 227
5.5.1 Volume Form......Page 229
5.5.2 Adjoint......Page 231
5.6.1 Gradient......Page 233
1.1 DENSITY......Page 264
1.2 INTEGRALS OF REAL-VALUED AND VECTOR FUNCTIONS......Page 266
2.1 DEFINITIONS......Page 268
2.2 SUBDIVISION......Page 270
3.1 MOMENTS AND PRODUCTS OF INERTIA......Page 272
3.2 INERTIA TENSOR......Page 274
4.1 MOMENT OF INERTIA ABOUT AN AXIS......Page 277
4.2 EQUATION OF THE QUADRIC......Page 279
4.3 NATURE OF THE QUADRIC......Page 280
5.1 FUNDAMENTAL THEOREM ABOUT SYMMETRIC TENSORS......Page 281
5.2 EQUAL EIGENVALUES......Page 285
5.3 INERTIA ELLIPSOID AND PRINCIPAL AXES......Page 287
5.4 MATERIAL SYMMETRIES......Page 288
6. STEINER’S THEOREM......Page 289
7. EXERCISES......Page 292
1. NEWTON’S POSTULATES......Page 300
1.1 EXPERIMENTAL LAWS......Page 301
1.2 POSTULATES......Page 302
1.3 GALILEAN RELATIVITY AND INERTIAL FRAMES......Page 304
2.1 KINETIC DYNAM......Page 308
2.2 KINETIC ENERGY......Page 309
3.1 FIRST INTEGRALS OF A SYSTEM OF PARTICLES......Page 310
3.2.1 Linear Momentum Theorem......Page 311
3.2.3 Theorem of Motion of Mass Center......Page 312
3.2.4 Special Case of Rigid Bodies......Page 313
3.3.1 Angular Momentum Theorem......Page 317
3.3.2 Relation between Kinetic Dynam and Dynam of Forces......Page 318
3.3.3 Conservation of Angular Momentum......Page 319
3.3.4 Special Case of Rigid Bodies......Page 320
3.4.1 Kinetic Energy Theorem......Page 324
3.4.2 Special Case of Rigid Bodies......Page 327
4. EXERCISES......Page 331
1. LAGRANGIAN DYNAMICS......Page 339
1.1 HOLONOMIC AND SCLERONOMIC SYSTEMS......Page 340
1.2 D’ALEMBERT-LAGRANGE PRINCIPLE......Page 342
1.3.1 Lagrange’s Equations in the General Case......Page 344
1.3.2 Lagrange’s Equations for Conservative Forces......Page 348
1.3.3 Lagrange’s Equations with Undetermined Multipliers......Page 350
1.4 CONFIGURATION SPACE AND LAGRANGE’S EQUATIONS......Page 354
1.5 ADJOINT LAGRANGIAN AND FIRST INTEGRALS......Page 358
2. VARIATIONAL CALCULUS AND PRINCIPLES......Page 360
2.1.1 A Variational Problem and Variations......Page 361
2.1.2 Euler’s Equations......Page 364
2.2.1 Hamilton’s Postulate......Page 368
2.2.2 Hamilton’s Principle and Motion Equations......Page 369
2.3 JACOBI’S FORM OF THE PRINCIPLE OF LEAST ACTION......Page 371
3.1 ONE-PARAMETER GROUP OF DIFFEOMORPHISMS......Page 374
3.2 EULER-NOETHER THEOREM......Page 376
4. EXERCISES......Page 379
1. N –BODY PROBLEM AND CANONICAL EQUATIONS......Page 392
2.1 LEGENDRE TRANSFORMATION AND HAMILTONIAN......Page 396
2.2 CANONICAL EQUATIONS......Page 400
2.3 FIRST INTEGRALS AND CYCLIC COORDINATES......Page 403
2.4 LIOUVILLE’S THEOREM IN STATISTICAL MECHANICS......Page 405
3.1.1 Preservation of Canonical Form and Poisson Bracket......Page 408
3.1.2 Poisson Bracket and Symplectic Matrix......Page 410
3.1.3 Lagrange and Poisson Brackets......Page 414
3.2 CANONICAL TRANSFORMATION......Page 415
3.2.1 Canonical Transformations and Brackets......Page 416
3.2.2 Canonical Transformations and Generating Functions......Page 418
4.1 HAMILTON-JACOBI EQUATION AND JACOBI THEOREM......Page 424
4.2 SEPARABILITY......Page 428
5. EXERCISES......Page 438
BIBLIOGRAPHY......Page 452
I N D E X......Page 453