Classical Mechanics (eBook)

Classical Mechanics 1
Preface 4
Contents 7
Chapter 1 Sketch of Lagrangian Formalism 11
1.1 Newton's Equation 11
1.2 Galilean Transformations: Principle of Galilean Relativity 18
1.3 Poincaré and Lorentz Transformations: The Principle of Special Relativity 23
1.4 Principle of Least Action 33
1.5 Variational Analysis 34
1.6 Generalized Coordinates, Coordinate Transformations and Symmetries of an Action 39
1.7 Examples of Continuous (Field) Systems 46
1.8 Action of a Constrained System: The Recipe 54
1.9 Action of a Constrained System: Justification of the Recipe 61
1.10 Description of Constrained System by Singular Action 62
1.11 Kinetic Versus Potential Energy: Forceless Mechanics of Hertz 64
1.12 Electromagnetic Field in Lagrangian Formalism 66
1.12.1 Maxwell Equations 66
1.12.2 Nonsingular Lagrangian Action of Electrodynamics 69
1.12.3 Manifestly Poincaré-Invariant Formulation in Terms of a Singular Lagrangian Action 73
1.12.4 Notion of Local (Gauge) Symmetry 75
1.12.5 Lorentz Transformations of Three-Dimensional Potential: Role of Gauge Symmetry 78
1.12.6 Relativistic Particle on Electromagnetic Background 79
1.12.7 Poincaré Transformations of Electric and Magnetic Fields 82
Chapter 2 Hamiltonian Formalism 86
2.1 Derivation of Hamiltonian Equations 86
2.1.1 Preliminaries 86
2.1.2 From Lagrangian to Hamiltonian Equations 88
2.1.3 Short Prescription for Hamiltonization Procedure, Physical Interpretation of Hamiltonian 92
2.1.4 Inverse Problem: From Hamiltonian to Lagrangian Formulation 94
2.2 Poisson Bracket and Symplectic Matrix 94
2.3 General Solution to Hamiltonian Equations 96
2.4 Picture of Motion in Phase Space 100
2.5 Conserved Quantities and the Poisson Bracket 102
2.6 Phase Space Transformations and Hamiltonian Equations 105
2.7 Definition of Canonical Transformation 109
2.8 Generalized Hamiltonian Equations: Example of Non-canonical Poisson Bracket 111
2.9 Hamiltonian Action Functional 115
2.10 Schrödinger Equation as the Hamiltonian System 116
2.10.1 Lagrangian Action Associated with the Schrödinger Equation 117
2.10.2 Probability as a Conserved Charge Via the Noether Theorem 120
2.11 Hamiltonization Procedure in Terms of First-Order Action Functional 122
2.12 Hamiltonization of a Theory with Higher-Order Derivatives 123
2.12.1 First-Order Trick 123
2.12.2 Ostrogradsky Method 125
Chapter 3 Canonical Transformations of Two-Dimensional Phase Space 128
3.1 Time-Independent Canonical Transformations 128
3.1.1 Time-Independent Canonical Transformations and Symplectic Matrix 128
3.1.2 Generating Function 130
3.2 Time-Dependent Canonical Transformations 132
3.2.1 Canonical Transformations and Symplectic Matrix 132
3.2.2 Generating Function 134
Chapter 4 Properties of Canonical Transformations 136
4.1 Invariance of the Poisson Bracket (Symplectic Matrix) 137
4.2 Infinitesimal Canonical Transformations: Hamiltonian as a Generator of Evolution 142
4.3 Generating Function of Canonical Transformation 145
4.3.1 Free Canonical Transformation and Its Function F(q', p', ) 145
4.3.2 Generating Function S(q, q', ) 146
4.4 Examples of Canonical Transformations 149
4.4.1 Evolution as a Canonical Transformation: Invariance of Phase-Space Volume 149
4.4.2 Canonical Transformations in Perturbation Theory 152
4.4.3 Coordinates Adjusted to a Surface 153
4.5 Transformation Properties of the Hamiltonian Action 154
4.6 Summary: Equivalent Definitions for Canonical Transformation 155
4.7 Hamilton--Jacobi Equation 156
4.8 Action Functional as a Generating Function of Evolution 160
Chapter 5 Integral Invariants 163
5.1 Poincaré-Cartan Integral Invariant 163
5.1.1 Preliminaries 163
5.1.2 Line Integral of a Vector Field, Hamiltonian Action, Poincaré-Cartan and Poincaré Integral Invariants 165
5.1.3 Invariance of the Poincaré--Cartan Integral 167
5.2 Universal Integral Invariant of Poincaré 170
Chapter 6 Potential Motion in a Geometric Setting 175
6.1 Analysis of Trajectories and the Principle of Maupertuis 175
6.1.1 Trajectory: Separation of Kinematics from Dynamics 176
6.1.2 Equations for Trajectory in the Hamiltonian Formulation 178
6.1.3 The Principle of Maupertuis for Trajectories 179
6.1.4 Lagrangian Action for Trajectories 180
6.2 Description of a Potential Motion in Terms of a Pair of Riemann Spaces 182
6.3 Some Notions of Riemann Geometry 186
6.3.1 Riemann Space 186
6.3.2 Covariant Derivative and Riemann Connection 191
6.3.3 Parallel Transport: Notions of Covariance and Coordinate Independence 193
6.4 Definition of Covariant Derivative Through Parallel Transport: Formal Solution to the Parallel Transport Equation 197
6.5 The Geodesic Line and Its Reparametrization Covariant Equation 199
6.6 Example: A Surface Embedded in Euclidean Space 201
6.7 Shortest Line and Geodesic Line: One More Example of a Singular Action 204
6.8 Formal Geometrization of Mechanics 208
Chapter 7 Transformations, Symmetries and Noether Theorem 211
7.1 The Notion of Invariant Action Functional 211
7.2 Coordinate Transformation, Induced Transformation of Functions and Symmetries of an Action 214
7.3 Examples of Invariant Actions, Galileo Group 219
7.4 Poincaré Group, Relativistic Particle 222
7.5 Symmetries of Equations of Motion 223
7.6 Noether Theorem 226
7.7 Infinitesimal Symmetries 228
7.8 Discussion of the Noether Theorem 231
7.9 Use of Noether Charges for Reduction of the Order of Equations of Motion 232
7.10 Examples 233
7.11 Symmetries of Hamiltonian Action 236
7.11.1 Infinitesimal Symmetries Given by Canonical Transformations 236
7.11.2 Structure of Infinitesimal Symmetry of a General Form 238
7.11.3 Hamiltonian Versus Lagrangian Global Symmetry 242
Chapter 8 Hamiltonian Formalism for Singular Theories 245
8.1 Hamiltonization of a Singular Theory: The Recipe 246
8.1.1 Two Basic Examples 246
8.1.2 Dirac Procedure 250
8.2 Justification of the Hamiltonization Recipe 255
8.2.1 Configuration-Velocity Space 255
8.2.2 Hamiltonization 257
8.2.3 Comparison with the Dirac Recipe 260
8.3 Classification of Constraints 262
8.4 Comment on the Physical Interpretation of a Singular Theory 263
8.5 Theory with Second-Class Constraints: Dirac Bracket 267
8.6 Examples of Theories with Second-Class Constraints 270
8.6.1 Mechanics with Kinematic Constraints 270
8.6.2 Singular Lagrangian Action Underlying the Schrödinger Equation 272
8.7 Examples of Theories with First-Class Constraints 274
8.7.1 Electrodynamics 274
8.7.2 Semiclassical Model for Description of Non Relativistic Spin 276
8.8 Local Symmetries and Constraints 282
8.9 Local Symmetry Does Not Imply a Conserved Charge 289
8.10 Formalism of Extended Lagrangian 289
8.11 Local Symmetries of the Extended Lagrangian: Dirac Conjecture 294
8.12 Local Symmetries of the Initial Lagrangian 298
8.13 Conversion of Second-Class Constraints by Deformation of Lagrangian Local Symmetries 301
8.13.1 Conversion in a Theory with Hidden SO(1, 4) Global Symmetry 304
8.13.2 Classical Mechanics Subject to Kinematic Constraints as a Gauge Theory 306
8.13.3 Conversion in Maxwell--Proca Lagrangian for Massive Vector Field 309
Bibliography 311
Index 313