Mathematical Sciences: Dynamics of Hamiltonian Systems

9401740马瑟·马瑟打算继续他在哈密顿系统动力学方面的工作。在过去，他的大部分工作都是关于两个自由度的面积保持映射和自治系统的动力学。最近，在他最近的论文《连接轨道的变分构造》中，他将这项工作的一部分扩展到了更多的自由度。然而，这篇论文只是在这个方向上的一个开始，在这个意义上，在更一般的假设下应该可以得到类似的结果。在更一般的假设下证明极小集之间存在连通轨道将是Mather研究的主要部分。作为这样一个结果的一个例子，考虑平面中的限制性三体问题，即具有牛顿引力势的力学问题，两个质量物体和一个零质量物体，都在平面中。在这两个大质量天体绕着对方的圆形轨道运动的情况下，莫泽在60年代初的S证明，如果零质量天体在接近圆形的轨道上运动，距离它们很远，那么它不会逃逸到无限远。因此，一个人就有了稳定。这是KAM理论的一部分。相反，如果两个大质量天体在椭圆轨道上相互运动，就不会有这样的结果。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，还有一些主要障碍需要克服。***

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. ***