经典n体问题无碰撞同变极小元的存在性
On the Existence of Collisionless Equivariant Minimizers for the Classical n-body Problem
摘要：
我们表明，最小化的拉格朗日作用功能的适当类的对称回路产生无碰撞的周期轨道的n体问题，只要满足一些简单的条件的对称群。更确切地说，我们给出了n体问题（具有齐次α的势，α>0）的环空间的对称群G的一个相当一般的条件，该条件确保拉格朗日作用量对G-等变环空间的限制是强制性的，并且其极小值是无碰撞的，没有任何强力假设。许多已知的周期轨道都可以通过这个结果得到证明，并且通过适当地选择G.
英文摘要：
We show that the minimization of the Lagrangian action functional on suitable classes of symmetric loops yields collisionless periodic orbits of the n-body problem, provided that some simple conditions on the symmetry group are satisfied. More precisely, we give a fairly general condition on symmetry groups G of the loop space for the n-body problem (with potential of homogeneous degree alpha, with alpha>0) which ensures that the restriction of the Lagrangian action to the space of G-equivariant loops is coercive and its minimizers are collisionless, without any strong force assumption. Many of the already known periodic orbits can be proved to exist by this result, and several new orbits are found with some appropriate choice of G.
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