Phase space, Hamilton’s equatios and their connection with Lagrange’s equations. Hamilton’s least action principle, Poisson brackets, integrals of motion.
Hamilton’s formalism, symplectic structure, Hamilton’s least action principle in symplectic structure, Poisson’s theorem.
- Canonical transformations, generating functions, criteria of canonical transformations.
- Symplectic matrices, the symplectic group, the eigenvalues of a syplectic matrix.
- Continuous families of canonical transformations, infinitesimal canonical transformations. Symmetries and their connection to integrals of motion, Noether’s theorem.
- Equillibrium points and their stability. Liouville’ theorem. Poincare’s theoem on repeatability.
- Hamilton-Jacobi equation. Hamilton-Jacobi for autonomous systems and separable systems. Integrable systems.
- Liouville’s integrability. Arnold-Liouville theorem for integrable systems and their topology.
- Action-angle variables in one degree of freedom Hamiltonian systems. Action-angle variables in n degrees of freedom Hamiltonian systems
- Poincare map in autonomous Hamiltonian systems. The twist map.
- Non itegrable Hamiltonian systems, systems near integrability. K.A.M. theorem
- Perturbed twist maps, Poincare-Birkhoff theorem.
- Homoclinic points and chaos.
- Summary of the course and extra excercises.