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

TL;DR

This paper proves the existence of minimal expansive solutions with prescribed asymptotic behavior in the classical N-body problem, including hyperbolic, parabolic, and hyperbolic-parabolic solutions, using a novel variational approach.

Contribution

It introduces a new minimization strategy for the action functional that confirms known solutions and establishes the existence of hyperbolic-parabolic solutions for the first time.

Findings

Confirmed existence of hyperbolic and parabolic solutions.

Proved existence of hyperbolic-parabolic solutions with prescribed asymptotics.

Provided detailed growth descriptions of these solutions.

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.