Lagrangian & Hamiltonian dynamics

553 Pages·2018·39.7 MB·English

by  Mann

LAGRANGIAN & HAMILTONIAN DYNAMICS

Lagrangian & Hamiltonian Dynamics
Peter Mann
UniversityofStAndrews
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3
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Contents
Preface xiii
PART I NEWTONIAN MECHANICS
1 Newton’s Three Laws 3
1.1 Phase Space 7
1.2 Systems of Particles 8
1.3 The N-body Problem 11
Chapter summary 13
2 Energy and Work 15
Chapter summary 22
3 Introductory Rotational Dynamics 24
Chapter summary 33
4 The Harmonic Oscillator 34
Chapter summary 38
5 Wave Mechanics & Elements of Mathematical Physics 44
PART II LAGRANGIAN MECHANICS
6 Coordinates & Constraints 55
Chapter summary 59
7 The Stationary Action Principle 61
7.1 The Inverse Problem 70
7.2 Higher-Order Theories & the Ostrogradsky Equation 72
7.3 The Second Variation 73
7.4 Functions & Functionals 74
7.5 Boundary Conditions 76
7.6 Variations 78
7.7 Weierstrass-Erdmann Conditions for Broken Extremals 79
7.8 Hamilton-Suslov Principle 79
Chapter summary 80
8 Constrained Lagrangian Mechanics 89
8.1 Holonomic Constraints 89
viii Contents
8.2 Non-Holonomic Constraints 93
Chapter summary 96
9 Point Transformations in Lagrangian Mechanics 100
Chapter summary 103
10 The Jacobi Energy Function 107
Chapter summary 112
11 Symmetries & Lagrangian-Hamilton-Jacobi Theory 115
11.1 Noether’s Theorem 115
11.2 Gauge Theory 120
11.3 Isotropic Symmetries 122
11.4 Caratheodory-Hamilton-Jacobi theory 123
Chapter summary 124
12 Near-Equilibrium Oscillations 130
12.1 Normal Modes 137
Chapter summary 140
13 Virtual Work & d’Alembert’s Principle 147
13.1 Gauss’s Least Constraint & Jourdain’s Principle 153
13.2 The Gibbs-Appell Equations 156
Chapter summary 158
PART III CANONICAL MECHANICS
14 The Hamiltonian & Phase Space 167
Chapter summary 172
15 Hamilton’s Principle in Phase Space 174
Chapter summary 178
16 Hamilton’s Equations & Routhian Reduction 179
16.1 Phase Space Conservation Laws 181
16.2 Routhian Mechanics 183
17 Poisson Brackets & Angular Momentum 190
17.1 Poisson Brackets & Angular Momenta 195
17.2 Poisson Brackets & Symmetries 197
Chapter summary 200
18 Canonical & Gauge Transformations 202
18.1 Canonical Transformations I 202
18.2 Canonical Transformations II 206
Contents ix
18.3 Infinitesimal Canonical Transformations 211
Chapter summary 214
19 Hamilton-Jacobi Theory 217
19.1 Hamilton-Jacobi Theory I 217
19.2 Hamilton-Jacobi Theory II 224
Chapter summary 229
20 Liouville’s Theorem & Classical Statistical Mechanics 237
20.1 Liouville’s Theorem & the Classical Propagator 237
20.2 Koopman-von Neumann Theory 244
20.3 Classical Statistical Mechanics 246
20.4 Symplectic Integrators 255
Chapter summary 259
21 Constrained Hamiltonian Dynamics 267
Chapter summary 274
22 Autonomous Geometrical Mechanics 277
22.1 A Coordinate-Free Picture 284
22.2 Poisson Manifolds & Symplectic Reduction 291
22.3 Geometrical Lagrangian Mechanics 296
22.4 Elements of Constrained Geometry 300
Chapter summary 303
23 The Structure of Phase Space 309
23.1 Time-Dependent Geometrical Mechanics 313
23.2 Picturing Phase Space 319
Chapter summary 322
24 Near-Integrable Systems 325
24.1 Canonical Perturbation Theory 325
24.2 KAM Theory & Elements of Chaos 333
PART IV CLASSICAL FIELD THEORY
25 Lagrangian Field Theory 345
Chapter summary 350
26 Hamiltonian Field Theory 353
27 Classical Electromagnetism 357
Chapter summary 365