We develop an index theory for parabolic and collision solutions to the classical n-body problem and we prove sufficient conditions for the finiteness of the spectral index valid in a large class of trajectories ending with a total collapse or expanding with vanishing limiting velocities. Both problems suffer from a lack of compactness and can be brought in a similar form of a Lagrangian System on the half time line by a regularising change of coordinates which preserve the Lagrangian structure. We then introduce a Maslov-type index which is suitable to capture the asymptotic nature of these trajectories as half-clinic orbits: by taking into account the underlying Hamiltonian structure we define the appropriate notion of geometric index for this class of solutions and we develop the relative index theory.

中文翻译：

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奇异势下渐近运动的指数理论

我们为经典 n 体问题的抛物线和碰撞解决方案开发了一个指数理论，并且我们证明了光谱指数的有限性的充分条件在以完全坍缩或以消失的极限速度扩展结束的一大类轨迹中有效。这两个问题都缺乏紧凑性，并且可以通过保持拉格朗日结构的坐标的正则化变化在半时间线上以类似的拉格朗日系统形式出现。然后，我们引入了适合将这些轨迹的渐近性质捕获为半临床轨道的 Maslov 型指数：通过考虑潜在的哈密顿结构，我们为此类解决方案定义了适当的几何指数概念，​​并开发了相对指数理论。