哈密顿动力系统周期轨道的多重性与稳定性研究

This project involves several subjects such as Hamiltonian system, differential geometry, mathematical physics and so on. Based on our systematic, innovative ideas and basis in the iteration theory of Maslov-type index during these years, we will establish the quasi-periodicity, common index quasi-periodicity and the symmetry of common index jump for iterated indices of periodic orbits. Then these properties will be further developed at the relative homology group level, which will be applied to deal with some difficulties such as the loss of index monotonicity, non-convexity of manifolds and non-reversibility of metrics. As applications, we will consider the following problems：the multiplicity of closed geodesics on every compact, simply-connected manifold; the sharp lower bound for the number of closed geodesics on every non-degenerate compact simply-connected manifold; the better lower bound for the number of closed geodesics under the pinching condition of the flag curvature; the multiplicity and stability of (symmetric) closed characteristics on (symmetric) compact convex or star-shaped hypersurfaces on $R^2n$. In addition, we also consider the counting problem of central configurations and the linear stability of elliptic periodic orbits in the N-body problem. These topics have extensive physical background and important theoretical significance. In recent years, researches in these fields are very active. The propose of this project is based on our solid background in the study on these topics and results we obtained, which have caused concern and interests of researchers in these fields.

本课题涉及哈密顿系统、微分几何与数学物理等多个学科。基于本课题组在Maslov-型指标迭代理论方面多年的系统的创新性思想和已有基础，我们将建立周期轨道的迭代指标的拟周期性、公共指标拟周期性、公共指标跳跃的对称性等性质，并把这些性质发展到同调上，用于处理迭代指标缺乏单调性、流形缺乏凸性、度量缺乏可逆性等带来的困难。作为应用，我们考虑如下的问题：任意紧单连通流形上闭测地线的多重性；任意非退化紧单连通流形上闭测地线条数的最优下界；旗曲率两边夹条件下闭测地线条数的更好下界；2n维欧式空间中（对称）紧凸或星型超曲面上（对称）闭特征的多重性和稳定性。另外，我们也将考虑N-体问题中的中心构型的计数和椭圆周期轨道的线性稳定性问题。这些问题具有广泛的物理背景和重要的理论意义。近年来国际上该领域的研究十分活跃。本项目的提出是基于我们在此课题研究中已具备的扎实的基础和做出的引起国际同行关注和感兴趣的研究成果。

本项目研究的科学问题涉及微分几何、哈密顿系统与天体力学等方向的一些基本问题，具有较强的物理学背景，是近年来国际研究的热点。项目主要的研究内容包括：流形上的闭测地线、哈密顿系统中给定能量面上的闭特征、辛道路指标的迭代理论，N-体问题中的椭圆周期解的线性稳定性。目前，对上述问题的研究存在一定的困难，本项目的特色和优势是指标迭代理论的进一步发展和应用。项目组成员在包括《Arch. Rational Mech. Anal.》、《Ann.I.H.Poincare-AN》、《Calc. Var. PDE》、《Nonlinearity》、《J. Differential Equations》等国际数学知名期刊上发表SCI论文16篇。主要成果包括：（1）在非退化和弱的指标条件下，证明了2n维欧式空间的紧星型超曲面上的闭特征的条数达到最优的下界；（2）证明了2n维欧式空间中的任意动力凸星型超曲面上至少存在n条闭特征（这里n=3或4）的长期多重性猜想；（3）建立了有界区间上线性哈密顿系统针对拉格朗日边值的Hill-型公式；（4）考虑了n体问题的椭圆相对平衡解、欧拉-Moulton共形解的线性稳定性，得到了其线性稳定与不稳定区域的分歧图;（5）把Krein-Lyubarski定理从解析线性推广到光滑非线性的情形等。本项目取得的成果引起了国际同行的关注和本质引用。项目执行期间，项目成员获得中国数学会第十三届华罗庚数学奖1项，获得2017年天津市数学会青年学术奖一等奖1项，培养博士后与研究生14名，举办国际国内学术会议4次，召集国际会议小组论坛1次，出国访问6次，参加国际会议7次，国际国内会议做邀请报告19次等。