Mathematical Sciences: Dynamics of Hamiltonian Systems

[401740]马瑟打算继续他在哈密顿系统动力学方面的工作。过去，他的大部分工作都是在两个自由度的区域保护映射和自治系统的动力学上。最近，在他最近的论文《连接轨道的变分构造》中，他将部分工作扩展到更大的自由度。然而，这篇论文只是在这个方向上的一个开始，从某种意义上说，在更一般的假设下应该有可能获得类似的结果。在更一般的假设下证明最小集之间的连接轨道的存在将是马瑟研究的主要部分。作为一个例子，考虑平面上的受限三体问题，即具有牛顿引力势的机械问题，两个有质量的物体和一个零质量的物体，都在平面上。在两个大质量物体在圆轨道上相互运动的情况下，莫泽在60年代早期指出，如果零质量物体在离大质量物体很远的近似圆轨道上运动，那么它不会逃逸到无穷大。这样就有了稳定性。这是KAM理论的一部分。相比之下，如果大质量天体在椭圆轨道上相互运动，则没有这样的结果。马瑟的方法可能会证明这样的结果是错误的。然而，在实施这一计划之前，还需要克服一些重大障碍。＊＊＊

9401740 Mather Mather intends to continue his work on dynamics of Hamiltonian systems. In the past, much of his work has been on dynamics of area-preserving mappings and autonomous systems in two degrees of freedom. More recently, he has extended part of this work to more degrees of freedom, in his recent paper, "Variational Construction of Connecting Orbits." However, this paper is only a beginning in this direction, in the sense that it should be possible to obtain similar results under much more general hypotheses. Proving the existence of connecting orbits between minimal sets under more general hypotheses will be a major part of Mather's research. As an example of what one might do with such results, consider the restricted three-body problem in the plane, i.e. the mechanical problem with a Newtonian gravitational potential, two massive bodies and one body of zero mass, all in the plane. In the case that the two massive bodies move about one another in circular orbits, Moser showed, in the early 60's, that if the zero mass body moves in an approximately circular orbit a long distance from the massive bodies, then it does not escape to infinity. Thus one has stability. This is part of KAM theory. In contrast, if the massive bodies move about one another in elliptical orbits, no such result is known. Mather's methods might lead to a proof that such a result is false. There are, however, major hurdles to be overcome before this program can be carried out. ***