Action-angle variables and adiabatic invariance in Hamiltonian systems

This thesis contains two main projects.

In the first project (Chapter 3), we consider a slow-fast Hamiltonian system with one fast angular variable (a fast phase) whose frequency vanishes on some surface in the space of slow variables (a resonant surface). Trajectories of the system averaged over the fast phase cross the resonant surface. The fast phase makes ∼1/ε turns before arrival at the resonant surface (ε is a small parameter of the problem). An asymptotic formula for the value of the phase at the arrival to the resonance was previously derived in the study of charged particle dynamics based on heuristic considerations without accuracy estimates. We provide a rigorous derivation of this formula and prove that its accuracy is O(√ε), up to a logarithmic correction. This estimate for the accuracy is optimal.

In the second project (Chapter 4), we consider two integrable Hamiltonian systems with spherical singularities and identical action variables. Existing studies have proven that action variables are preserved under fiberwise symplectomorphisms and that symplectomorphic systems have identical action variables. However, do non-symplectomorphic spherical singularities with identical actions exist? We develop methods to construct two integrable systems with identical actions in various situations, check if they are symplectomorphic, and provide a positive answer to this question with an example.