## A Course in Mathematical Physics 1 and 2: Classical Dynamical Systems and Classical Field Theory, Volumes 1-2The last decade has seen a considerable renaissance in the realm of classical dynamical systems, and many things that may have appeared mathematically overly sophisticated at the time of the first appearance of this textbook have since become the everyday tools of working physicists. This new edition is intended to take this development into account. I have also tried to make the book more readable and to eradicate errors. Since the first edition already contained plenty of material for a one semester course, new material was added only when some of the original could be dropped or simplified. Even so, it was necessary to expand the chap ter with the proof of the K-A-M Theorem to make allowances for the cur rent trend in physics. This involved not only the use of more refined mathe matical tools, but also a reevaluation of the word "fundamental. " What was earlier dismissed as a grubby calculation is now seen as the consequence of a deep principle. Even Kepler's laws, which determine the radii of the planetary orbits, and which used to be passed over in silence as mystical nonsense, seem to point the way to a truth unattainable by superficial observation: The ratios of the radii of Platonic solids to the radii of inscribed Platonic solids are irrational, but satisfy algebraic equations of lower order. |

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### Contents

Introduction | 1 |

Analysis on Manifolds | 8 |

Hamiltonian Systems | 84 |

Copyright | |

4 other sections not shown

### Common terms and phrases

algebra asymptotic bijection boundary bundle calculate canonical transformation Cauchy surface causal charge chart compact components constants of motion convergence coordinate system cosh covariant curvature curve defined Definition density diffeomorphism differential eigenvalues Einstein's equations electromagnetic energy energy-momentum equations of motion example exist exterior factor Figure finite flow force frequency gauge geodesic global gravitational field Green function Hamiltonian Hence implies independent infinite infinity integral interior product Killing vector fields Lagrangian Lie derivative linear Lorentz Lorentz transformation manifold mapping matrix Maxwell's equations metric Minkowski space natural basis orbits orthogonal basis oscillator particle phase space Poisson brackets potential Problem pseudo-Riemannian radiation region relativistic Remarks restriction Riemannian rotating satisfies scalar product Schwarzschild Schwarzschild metric Show singularity sinh solution spacelike structure submanifold subset surface tangent tensor field term theorem theory time-evolution topology trajectories vanish variables velocity wave zero