## Abstract

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.

Original language | English (US) |
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Article number | 107230 |

Journal | Advances in Mathematics |

Volume | 370 |

DOIs | |

State | Published - Aug 26 2020 |

## Keywords

- Colliding trajectories
- Homothetic orbits
- Index theory
- Maslov index
- Parabolic motions
- Spectral flow

## ASJC Scopus subject areas

- General Mathematics