Complex Dynamics of Hamiltonian Systems and Applications

The course aims to develop the basic concepts and methods of Hamiltonian description for the study of complex and chaotic dynamics of systems with applications in physics and technology. Content: Formalism and methods of Hamilton mechanics: Canonical variables and symplectic formalism. Canonical transformations and generating functions. Poisson bracket formalism. Infinitesimal canonical transformations, symmetries, and invariant quantities. Hamilton-Jacobi theory, integrability, and separability. Action-angle variables and adiabatic invariants. Near-integrable systems and Canonical Perturbation Theory. Complex dynamics and chaos: Resonances and the problem of small denominators. Poincaré maps and surfaces of section. Kolmogorov-Arnold-Moser theorem and Hamiltonian chaos. Criteria for transition to chaos. Stochastic approximation and diffusion in phase space. Applications: Forced, parametrically excited, and coupled nonlinear oscillators. Non-linear interaction of charged particles with electromagnetic fields.