This article is based on a course the author gave in fall of 2018 at UC Berkeley, in connection with the MSRI program Hamiltonian systems, from topology to applications through analysis. In this article we explore the connection

between the Hamiltonian ODEs and Hamilton–Jacobi PDEs, and give an overview of some of the existing techniques for the question of homogenization. We also discuss stochastic formulations of several classical problems in symplectic geometry.

Hamiltonian systems of ordinary differential equations appear in celestial mechanics to describe the motion of planets. They are also used in statistical mechanics to model the dynamics of particles in a fluid, gas or many other microscopic models. It was known to Liouville that the flow of a Hamiltonian system preserves the volume. Poincaré observed that the Hamiltonian flows are symplectic; they preserve certain symplectic area of two dimensional surfaces. Various symplectic rigidity phenomena offer ways to take advantage of the symplecticity of Hamiltonian flows.

Writing q and p for the position and momentum coordinates respectively, a

Hamiltonian function H(q, p) represents the total energy associated with the pair (q, p). We regard a Hamiltonian system associated with H completely integrable

if there exists a symplectic change of coordinates (q, p) →(Q, P), such that our Hamiltonian system in new coordinates is still Hamiltonian system that is now associated with a Hamiltonian function H(P). For completely integrable systems the coordinates of P = P(q, p) are conserved and the set of (q, p) at which P(q, p) takes a fixed vector is an invariant set for the flow of our system. These invariant sets are homeomorphic to tori in many classical examples of completely integrable systems. According to Kolmogorov–Arnold–Moser (KAM) theory, many of the invariant tori survive when a completely integrable system is slightly perturbed. Aubry–Mather theory constructs a family of invariant sets provided that the Hamiltonian function is convex in the momentum variable. These invariant sets lie on the graph of the gradient of certain scalar-valued functions. A. Fathi [10] uses viscosity solutions of the Hamilton–Jacobi PDE associated with the Hamiltonian function H to construct Aubry–Mather invariant measures; see also [3]. Recently there have been several interesting works to understand the connection between Aubry–Mather theory and symplectic topology. The hope is to use tools from symplectic topology to construct interesting invariant sets/measures for Hamiltonian systems associated with nonconvex Hamiltonian functions.