On the existence of minimal expansive solutions to the $N$-body problem

Davide Polimeni; Susanna Terracini

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arXiv · 2310.06360

On the existence of minimal expansive solutions to the $N$-body problem

Davide Polimeni·Susanna Terracini

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Abstract

We deal, for the classical $N$-body problem, with the existence of action minimizing half entire expansive solutions with prescribed asymptotic direction and initial configuration of the bodies. We tackle the cases of hyperbolic, hyperbolic-parabolic and parabolic arcs in a unitary manner. Our approach is based on the minimization of a renormalized Lagrangian action, on a suitable functional space. With this new strategy, we are able to confirm the already-known results of the existence of both hyperbolic and parabolic solutions, and we prove for the first time the existence of hyperbolic-parabolic solutions for any prescribed asymptotic expansion in a suitable class. Associated with each element of this class we find a viscosity solution of the Hamilton-Jacobi equation as a linear correction of the value function. Besides, we also manage to give a better description of the growth of parabolic and hyperbolic-parabolic solutions.

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Davide Polimeni, Susanna Terracini. 2023-10-10. On the existence of minimal expansive solutions to the $N$-body problem. https://arxiv.org/abs/2310.06360

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