On the existence of minimal expansive solutions to the \(N\)-body problem (Q6628961)

In this paper, the authors address the existence of half-entire expansive solutions for the Newtonian \(N\)-body problem from a unified perspective. Their approach employs a global variational method based on a suitably renormalized Lagrangian action functional, as the standard Lagrangian is not integrable on the half-line.\N\NDrawing on \textit{J. Chazy}'s classification [Ann. Sci. Éc. Norm. Supér. (3) 39, 29--130 (1922; JFM 48.1074.04)], they provide a proof for the existence of hyperbolic, hyperbolic-parabolic, and parabolic motions. As a preliminary step, the authors revisit recent works by Maderna and Venturelli, examining the existence of half-hyperbolic and parabolic trajectories from a fresh perspective. In [Ann. Math. (2) 192, No. 2, 499--550 (2020; Zbl 1475.70018)], \textit{E. Maderna} and \textit{A. Venturelli} demonstrated the existence of hyperbolic motions for any prescribed limit shape, initial configuration of the bodies, and any positive energy value, relying on the construction of global viscosity solutions for the Hamilton-Jacobi equation. Similarly, in [\textit{E. Maderna} and \textit{A. Venturelli}, Arch. Ration. Mech. Anal. 194, No. 1, 283--313 (2009; Zbl 1253.70015)], they established the existence of parabolic arcs asymptotic to any prescribed normalized minimal central configuration, for arbitrary starting configurations. These solutions were derived as limits of sequences solving approximating two-point boundary problems.\N\NIn contrast to these works, the paper under review offers alternative and more concise proofs for the existence of hyperbolic and parabolic solutions within a unified framework. This framework is based on the application of the Direct Method in the Calculus of Variations to minimize the renormalized Lagrangian action functional associated with the problem. Furthermore, the authors extend their approach to establish the existence of hyperbolic-parabolic solutions for the \(N\)-body problem.