In many areas of mathematics and physics it is important to analyze the dynamical behavior of systems that conserve some important physical quantities such as energy, mass, or volume. These are called conservative systems, and one natural type of conservative system is a Hamiltonian system of finite or infinite dimension (named after W. Hamilton, who studied them in the 19th century). Hamiltonian systems are derived from classical mechanics and remain fundamental objects of study. It is critically important to understand the formation of instabilities of solutions for these systems. One of the goals of this proposal is to study dynamics of nearly integrable Hamiltonian systems, systems whose solutions can be approximated by explicit formulas, as well as their instabilities. This study will provide significant modern advances in understanding the classical problems, with many current applications, especially to astrophysics. The PI expects to make progress in this direction in part by extending previous successful work by him to higher dimensional systems.
The proposal will be focused around proving strong forms of Arnol'd diffusion for a variety of Hamiltonian systems and studying spectral rigidity for convex billiards (elastic collisions). The major topics of the proposal are: Arnol'd diffusion for convex Hamiltonian systems, existence of an almost dense orbit on an energy surface for nearly integrable Hamiltonian systems. The latter is a weak form of quasi-ergodic hypothesis. Another direction of our research is to prove stochastic and diffusive behavior for nearly integrable systems, where randomness is with respect to initial conditions. We also plan to study growth of Sobolev norms for Hamiltonian PDEs and extend our previous results on this topic.
