N-体问题中的周期轨道研究

Since A.Chenciner and R.Montgomery proved the existence of the remarkable figue eight orbit, variational methods have been widely applied into the study of N-body problem. The idea of proving the existence of periodic solution is to introduce some symmetric constraints on the path space to ensure the coercivity of action functional. If one can prove the minimizer to be collision-free, then it is standard to get the existence proof of the periodic solution. Here we will use a two point free boundary variational method to study the existence of periodic orbits. When order constraints are added to the boundary condition, it will become much more difficult to eliminate the collisions. We will focus on studying this kind of problems. We hope to find a way to deal with the possible collisions with order constarints at the boundary. We'll also study the properties of periodic orbits and try to give a more accurate description about the orbits..On the other hand, the stability of periodic orbits is also of great interest. Recently, there are many developments in Maslov-type index theory applying to the study of linear stability of perioidc solutions. Related mathematical tools are extened to Hamiltonian systems with symmetric group action. Depending on the new developments, we will also study the stability of related periodic orbits.

自从A.Chenciner和R.Montgomery通过对轨道引入对称性限制证明了8字形轨道的存在性以来，变分法在N-体问题的研究中得到了广泛的应用。周期轨道存在性证明的通常思路是通过对轨道或者边界条件加上对称性限制，从而保证泛函的强制性，然后排除可能发生的碰撞得到周期轨道的存在性证明。我们这里将利用两点自由边值的变分方法来研究周期轨道的存在性，在边界条件对质点具有顺序限制时，碰撞的排除变得十分困难。我们将会对这类问题进行深入的研究和讨论，希望找到处理它们的方法和规律。同时，我们也会对变分极小轨道的各种性质进行研究，试着给出一些周期轨道更精确的刻画。.另一方面，周期轨道的稳定性问题也是人们关注的热点。近些年来Maslov指标理论在周期轨道稳定性的研究方面有很多新的进展，相关的结论被推广到了具有对称群作用的哈密顿系统中，借助这些新的方法和工具，我们会对一些特殊周期轨道的稳定性进行研究。

近些年来变分法在N体问题的研究中得到了广泛应用，本项目在相关工作的基础之上进一步对N体问题周期轨道的相关内容进行研究。项目主要研究结果包括：（1）在Broucke-Henon轨道的存在性问题得到重要进展；（2）证明2n体问题在三维空间中的一类对称轨道的存在性；（3）得到3体问题在一定边界条件下变分极小轨道的部分几何性质。这些工作有助于我们进一步了解N体问题中的周期轨道及其性质，相关成果发表在《Calculus of Variations and Partial Differential Equations》、《Journal of Dynamics and Differential Equations》等期刊上。