Final Report Summary - SAW (Symplectic Aspects of Weak KAM theory.)
The goal of the project was to to study Symplectic Aspects of Weak kam theory.
One of the motivation for such a program was to understand to what extent this theory could possibly be extended to some classes of non convex Hamiltonian. Another motivation was to be able to use simultaneously on a given problem standard methods from Hamiltonian dynamics such as averaging and weak KAM methods.
In the first direction, a rather negative result was obtained: We proved that convex (or concave) Hamiltonians are the only ones for which methods closely similar to weak KAM theory can be applied.
In the second direction, we gave the first proof of the generic existence of Arnodl's diffusion in arbitrary dimension, precisely by mixing weak KAM and classical Hamiltonian thechniques.
One of the motivation for such a program was to understand to what extent this theory could possibly be extended to some classes of non convex Hamiltonian. Another motivation was to be able to use simultaneously on a given problem standard methods from Hamiltonian dynamics such as averaging and weak KAM methods.
In the first direction, a rather negative result was obtained: We proved that convex (or concave) Hamiltonians are the only ones for which methods closely similar to weak KAM theory can be applied.
In the second direction, we gave the first proof of the generic existence of Arnodl's diffusion in arbitrary dimension, precisely by mixing weak KAM and classical Hamiltonian thechniques.