Research In Hamiltonian Dynamics

夏教授建议继续他对哈密顿系统动力学的研究。哈密顿动力学系统中最重要的问题之一是典型系统的轨道是否稳定。所提出的研究将解决这些问题，研究主题包括Aubry-Mather理论、Arnold扩散、混沌行为和牛顿n元体问题。最终目的是证明高维典型的近可积哈密顿量在拓扑上是不稳定的。哈密顿动力系统模型许多系统起源于经典力学、天体力学和物理学。这些问题由来已久，可以追溯到庞加莱和伯克霍夫。人们经常问的非典型问题是：我们的太阳系是稳定的吗？我们的目标之一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最新进展，我们对典型系统的混沌性质有了更深入的了解。然而，除了一些特殊情况外，稳定性问题仍然是一个悬而未决的问题，无论是从理论角度还是在各种物理系统中的广泛应用，它都是最重要的研究领域之一。

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.