我们主要采取变分法和PDE方法结合的方法，研究高维哈密尔顿系统中的不稳定行为。我们的主要目标有：1.研究经典力学系统中无界轨道的通有存在性。这个问题来源于J．Mather关于经典力学系统无界轨道存在性的猜测和物理学中著名的费米粒子加速问题。我们拟通过以下两个步骤实现：(1)研究构型空间为n维环面时，由平坦度量定义的动能加通有势能（依赖时间）构成的力学系统中无界轨道的存在性。(2)进一步的，研究通有动能在通有势能（依赖时间）构成的力学系统中无界轨道的存在性。2.研究一般维度的辛映射或Hamilton系统的频率共振的椭圆不动点的稳定性问题。3.拟借助弱KAM理论，通过研究粘性解的稳定性问题来研究障碍函数的稳定性问题，后者是用Mather理论研究Arnold扩散最本质的困难所在。

We want to study the instability in high-dimensional Hamiltonian system through the method which combines variational approach with PDE method. The main goal including: 1. We shall study the generic existence of the unbounded orbits in classic mechanics system. This problem comes from J. Mather's conjecture about the existence of unbounded orbits in classic mechanics system and the Fermi particle acceleration problem in physics. To complete the task, we adopt the steps as follows: (1) We intend to prove the existence of the unbounded orbits in the mechanics system which comprizes the kinetic energy defined by flat metric and generic potential energy (dependent of time). (2) Furthermore, we intend to the existence of the unbounded orbits in the mechanics system which comprizes generic kinetic energy and generic potential energy (dependent of time). 2.We shall study the stability of elliptic fixed point with resonant frequency in the high-dimensional symplectic mapping or Hamiltonian system. (3) We want to study the stability of the barrier function by studying the corresponding viscosity solution and the former is the key point to study Arnold Diffusion by Mather Theory.