Research In Hamiltonian Dynamics

夏教授建议继续他对哈密顿系统动力学的研究。Hamilton代数系统中最重要的问题之一是典型系统的轨道是否稳定。 本研究将针对这些问题，研究主题包括Aubry-Mather理论、Arnold扩散、混沌行为和牛顿体问题。本文的最终目的是证明高维近似可积Hamilton算子是拓扑不稳定的，所提出的研究涉及Hamilton算子和Hamilton算子的稳定性问题。Hamilton动力系统是经典力学、天体力学和物理学中许多系统的模型。这些问题由来已久，可以追溯到庞加莱和伯克霍夫。人们经常问的一个非典型问题是：我们的太阳系稳定吗？我们的目标之一就是找到这个问题和一大类相关问题的答案。随着现代动力系统理论的发展，我们对典型系统的混沌性质有了更深入的了解。然而，稳定性问题除了在某些特殊情况下仍然是开放的，它是最重要的研究领域之一，无论是从理论的角度来看，它发现在各种物理系统的广泛应用。

Professor Xia proposes to continue his investigation of dynamics ofHamiltonian systems. One of the most important problems in Hamiltoniandynamical systems is whether orbits in typical systems are stable. Theproposed research will address these problems, with research topics such asAubry-Mather theory, Arnold diffusions, chaotic behaviors and Newtonian$n$-body problem. The ultimate goal is to show that typical near integrableHamiltonians in higher dimensions are topologically unstable.The proposed research concerns the stability problems in Hamiltoniandyanmics. Hamiltonian dynamical systems model many systems arisingfrom classical mechanics, celestial mechanics and physics. Theseproblems has a long history, going back to Poincare and Birkhoff. Atypical question one often asks is the following: Is our solar systemstable? One of our goals is to find an answer to this and a largeclass of related questions. With recent progresses in the theory ofmodern dynamical systems, we understand much better the chaotic natureof typical systems. However, the stability problem remains open exceptin some special cases and it is one of the most important areas ofstudy, from both theoretical point of view and wide applications itfinds in various physical systems.