Symmetric Solutions to Nonlinear Vectorial 2nd Order ODE's

TL;DR

This paper proves the existence of symmetric solutions, including even and odd solutions, for a broad class of second-order nonlinear vectorial differential systems, with applications to celestial mechanics.

Contribution

It establishes the existence of symmetric solutions for nonlinear vectorial second-order ODEs, including systems like the N-body problem, expanding understanding of their solution structure.

Findings

Existence of a continuum of symmetric solutions.

Presence of even solutions under general conditions.

Existence of odd solutions when f(y) is odd.

Abstract

It is proven that second-order vectorial nonlinear differential systems y''=f(y) , possess a continuum of symmetric solutions. They are shown to possess a continuum of even solutions. If f(y) is an odd function of y , then y''=f(y) is shown also to possess a continuum of odd solutions. The results apply to a significant family of second-order vectorial nonlinear differential systems that are not dissipative. This family of differential equations includes the celebrated N-body problem of celestial mechanics and other central force problems.