Symmetric periodic solutions in the generalized Sitnikov Problem with homotopy methods

TL;DR

This paper explores symmetric periodic solutions in a generalized Sitnikov problem involving multiple primaries, using homotopy methods and degree theory to prove the existence of infinitely many solutions with increasing periods.

Contribution

It introduces a novel application of homotopy and degree theory to establish the existence of symmetric periodic solutions in a generalized Sitnikov problem.

Findings

Existence of infinitely many even solutions

Existence of infinitely many anti-periodic solutions

Solutions have increasing periods

Abstract

The paper investigates a generalization of the classical Sitnikov problem, concentrating on the movement of a satellite along the Z-axis as it interacts with $n$ primary bodies in periodic motion. It establishes the existence of an infinite number of even and anti-periodic solutions with increasing periods. The proof employs the Leray-Schauder degree theory to trace the critical points of action functionals, using a homotopy from solutions when the primary bodies are transformed into circular orbits.