Preface ......................................................... v
1 Lagrangian dynamics .......................................... 1
1.1 Introduction ............................................ 1
1.2 Generalized coordinates and the Lagrangian .............. 2
1.3 Virtual work and generalized force ...................... 4
1.4 Conservative forces and the Lagrangian .................. 6
1.4.1 The central force problem in a plane .............. 7
1.5 Noether's theorem ....................................... 9
1.6 Velocity-dependent forces and potentials ............... 11
1.7 The Hamiltonian formulation ............................ 13
1.7.1 The spherical pendulum ........................... 15
1.8 Summary ................................................ 17
1.9 Sources and references ................................. 18
1.10 Problems ............................................... 18
2 Canonical transformations ................................... 23
2.1 Contact transformations ................................ 23
2.1.1 The harmonic oscillator: cracking a peanut with
a sledgehammer ................................... 26
2.2 The second generating function ......................... 27
2.3 Hamilton's principal function .......................... 28
2.3.1 The harmonic oscillator: again ................... 30
2.4 Hamilton's characteristic function ..................... 31
2.4.1 Examples ......................................... 32
2.5 Action-angle variables ................................. 33
2.5.1 The harmonic oscillator: yet again ............... 35
2.6 Summary ................................................ 37
2.7 Sources and references ................................. 38
2.8 Problems ............................................... 38
3 Abstract transformation theory .............................. 41
3.1 Notation ............................................... 41
3.2 Poisson brackets ....................................... 43
3.3 Liouville's theorem .................................... 46
3.4 Geometry in n dimensions: the hairy ball ............... 52
3.5 Summary ................................................ 56
3.5.1 Example: uncoupled oscillators .................. 56
3.5.2 Example: a particle in a box .................... 58
3.6 Problems ............................................... 59
3.7 Sources and references ................................. 60
4 Canonical perturbation theory ............................... 61
4.1 One-dimensional systems ................................ 61
4.2 Summary ................................................ 65
4.2.1 The simple pendulum .............................. 65
4.3 Many degrees of freedom ................................ 67
4.4 Problems ............................................... 69
4.5 Sources and references ................................. 69
5 Introduction to chaos ....................................... 71
5.1 The total failure of perturbation theory ............... 72
5.2 Fixed points and linearization ......................... 73
5.2.1 Two examples ..................................... 78
5.3 The Henon-Heiles oscillator ............................ 80
5.4 Discrete maps .......................................... 84
5.5 Linearized maps ........................................ 88
5.6 Lyapunov exponents ..................................... 90
5.7 The Poincare-Birkhoff theorem .......................... 91
5.8 All in a tangle ........................................ 96
5.9 An example: coupled pendulums .......................... 99
5.10 The KAM theorem: background ........................... 101
5.11 Statement of the theorem .............................. 108
5.11.1 Two conditions ................................. 108
5.12 Analysis .............................................. 110
5.13 Conclusion ............................................ 111
5.14 Sources and references ................................ 112
6 Computational projects ..................................... 113
6.1 The Henon-Heiles Hamiltonian .......................... 113
6.2 The orbit of Mercury .................................. 114
6.3 The standard map ...................................... 115
6.4 The swinging Atwood's machine ......................... 116
Appendix A Measure theory and the ergodic hypothesis ........ 119
Bibliography .................................................. 125
Index ......................................................... 127
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