In the Newtonian n-body problem for solutions with arbitrary energy, which start and end either at a total collision or a parabolic/hyperbolic infinity, we prove some basic results about their Morse and Maslov indices. Moreover for homothetic solutions with arbitrary energy, we give a simple and precise formula that relates the Morse indices of these homothetic solutions to the spectra of the normalized potential at the corresponding central configurations. Potentially these results could be useful in the application of non-action minimization methods in the Newtonian n-body problem.

中文翻译：

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牛顿n体问题的碰撞，抛物线和双曲解的索引理论

在具有任意能量的牛顿n体问题中，该问题以完全碰撞或抛物线/双曲线无穷大为起点和终点，我们证明了它们的Morse和Maslov指数的一些基本结果。此外，对于具有任意能量的同构解决方案，我们给出了一个简单而精确的公式，该公式将这些同构解决方案的莫尔斯指数与相应中心构型下的归一化势能谱相关。这些结果可能在非作用最小化方法在牛顿n体问题中的应用中可能是有用的。