哈密顿动力系统中的变分法应用分析  

Application of Variational Method in Hamiltonian Dynamical Systems

作　　者：张建路 ZHANG Jianlu(Hua Loo-Keng Key Laboratory of Mathematics,Academy of Mathematics and Systems Science,Chinese Academy of Sciences,Beijing 100190)

机构地区：[1]中国科学院数学与系统科学研究院,华罗庚数学重点实验室,北京100190

出　　处：《中国基础科学》2024年第3期57-62,共6页China Basic Science

基　　金：国家重点研发计划青年科学家项目(2022YFA1007500)。

摘　　要：拉格朗日变分法极大地推动了哈密顿动力系统的定性研究。这一方法主要利用变分观点的“极小性”来细化动力系统的不变集,使其参数化为依赖同调(上同调)群的层状结构。利用这一分层结构的良好特性,可以揭示系统全局动力学信息以及通有特征。近年来,这一思想延伸至偏微分方程和最优控制等领域,在许多实际问题中发挥了巨大的效用。全面地分析这一研究方向的前沿问题,尤其是中国学者取得的突出成果及优势领域。此外,简明地梳理变分法在哈密顿动力系统中应用的发展脉络,总结其中的成功经验,并对这一行业的发展前景进行明确的规划及展望。Lagrangian variational method boosts the qualitative research of Hamiltonian dynamical systems greatly.The motivation of this method is to make partitions of the invariant sets by the“variational minimum”,and parametrize them into a foliation with respect to the homology(cohomology)group.Making use of the fine properties of such a structure,the global dynamics and the generic features of invariant sets can be revealed.In these years this method has been applied to other subjects,e.g.the partial differential equations,optimal controls,etc.In some practical occasions,it also makes great effects.This paper analyses the front problems of this subject with details.It exhibits the achievements of Chinese experts and the associated advantages.Besides,it concisely summarizes the development of the variational method in Hamiltonian systems,and the useful experiences within it,and finally programs the new trend of this subject clearly.

关 键 词：哈密顿动力系统 Aubry Mather理论 拉格朗日变分法 哈密顿—雅克比方程 

分 类 号：O176[理学—数学]