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Morse index properties of colliding solutions to the $N$-body problem

Vivina Barutello·Simone Secchi
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Abstract

We study a singular Hamiltonian system with an $\al$-homogeneous potential that contains, as a particular case, the classical $N$--body problem. We introduce a variational Morse--like index for a class of collision solutions and, using the asymptotic estimates near collisions, we prove the non-minimality of some special classes of colliding trajectories under suitable spectral conditions provided $\al$ is sufficiently away from zero. We then prove some minimality results for small values of the parameter $\al$.

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Vivina Barutello, Simone Secchi. 2006-09-29. Morse index properties of colliding solutions to the $N$-body problem. https://arxiv.org/abs/math/0609837

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