Notes on Dynamical Systems

Preface
Chapter 1. Transformation Theory
1.1. Differential Equations and Vector Fields
1.2. Variational Principles, Hamiltonian Systems
1.3. Canonical Transformations
1.4. Hamilton-Jacobi Equations
1.5. Integrals and Group Actions
1.6. The SO(4) Symmetry of the Kepler Problem
1.7. Symplectic Manifolds
1.8. Hamiltonian Vector Fields on Symplectic Manifolds
Chapter 2. Periodic Orbits
2.1. Poincare's Perturbation Theory of Periodic Orbits
2.2. A Theorem by Lyapunov
2.3. A Theorem by E. Hopf
2.4. The Restricted 3-Body Problem
2.5. Reversible Systems
2.6. The Plane 3- and 4-Body Problems
2.7. Poincare-Birkhoff Fixed Point Theorem
2.8. Variations on the Fixed Point Theorems
2.9. The Billiard Ball Problem
2.10. A Theorem by Jacobowitz and Hartman
2.11. Closed Geodesies on a Riemannian Manifold
2.12. Periodic Orbits on a Convex Energy Surface
2.13. Periodic Orbits Having Prescribed Periods
Chapter 3. Integrable Hamiltonian Systems
3.1. A Theorem of Arnold and Jost
3.2. Delaunay Variables
3.3. Integrals via Asymptotics; the Stormer Problem
3.4. The Toda Lattice
3.5. Separation of Variables
3.6. Constrained Vector Fields
3.7. Isospectral Deformations
Bibliography