Some recent progress in geometric methods in the instability problem in Hamiltonian mechanics

Abstract

We discuss some geometric structures that lead to instability in Hamiltonian systems arbitrarily close to integrable. The structures covered in this report are joint work with A. Delshams, T. M. Seara and M. Gidea.

Arnold Diffusion in A Priori Chaotic Symplectic Maps

“…The criteria for the existence of trajectories of the energy that grows up to infinity are known for sufficiently large initial energies [10,24,29,30,44,45,80,81]. The results of the present paper can be used to establish the generic existence of orbits of unbounded energy for all possible values of initial energy.…”

“…where, for the derivatives in the right-hand side, we use the supremum norm taken over all (u, v,z) such that u,z ≤ δ, and C 0 (δ) is defined by (24). By (12)- (16), if δ is sufficiently small, then Q α < 1 uniformly for every element from X , independently of the value of k ≥ 0.…”

Variational construction of positive entropy invariant measures of Lagrangian systems and Arnold diffusion

“…Importantly, we do not require any non-degeneracy of the minima of S ω , for example we do not require that the second derivative of S ω at these minima is positive definite. As perhaps noted first by Angenent [1], such non-degeneracy would be equivalent to requiring that the stable and unstable manifolds (the "whiskers") of T ω intersect transversally, and would lead to the "geometric" approach to diffusion phenomena ( [29] and references therein). On the contrary, we allow the whiskers to intersect non-transversally, and construct possibly non-uniformly hyperbolic invariant sets.…”

Geometric approaches to the problem of instability in Hamiltonian systems. An informal presentation

“…The only goal of these lectures is to present a road map to the programs and to indicate the significant mileposts to be reached. Some similar expositions are [DDLLS00, DLS03,dlL06]. The present one incorporates some progress since the previous exposition were written.…”

“…One systematic way to make sense of the above [Lla04,dlL06] is to observe that the set ∪ x∈Λ {x} × M is a normally hyperbolic lamination for F 0 . See Appendix A.3.…”