本项目将利用辛拓扑领域中的辛齐次化理论来研究非凸Hamilton的动力学, 为建立非凸Hamilton系统的Aubry-Mather理论奠定初步的基础. 具体来讲, 我们将在非凸Hamilton系统中构建合适的变分原理, 证明Aubry-Mather集类型的不变集的存在性, 并构造这些不变集之间的连接轨道; 我们将研究非凸Hamilton系统的Hamilton-Jacobi方程的粘性解, 极小极大解和Hamilton系统的动力学之间的联系.

In this program we will study the dynamics for non-convex Hamiltonians by the theory of symplectic homogenization from the field of symplectic topology, and build a foundation for Aubry-Mather theory of non-convex Hamiltonian dynamical systems. More precisely, we will construct a suitable variational principle, then prove the existence of various kinds of invariant sets analogous to Aubry-Mather sets, and construct connecting orbits visiting these invariant sets; we will study the relationships among viscosity solutions, minmax sloutions of the associated Hamilton-Jacobi equations and the dynamics of the non-convex Hamiltonians.