Chaos in a generalized Euler's three-body problem

TL;DR

This paper investigates a generalized version of Euler's three-body problem with an inverse-square potential, revealing stable stationary orbits and chaotic dynamics, thereby demonstrating the system's nonintegrability.

Contribution

It introduces a higher-dimensional generalization of Euler's problem, identifies stable stationary orbits, and proves the presence of chaos, showing the system is nonintegrable.

Findings

Existence of stable stationary orbits

Presence of stable bound chaotic orbits

The system is nonintegrable

Abstract

Euler's three-body problem is the problem of solving for the motion of a particle moving in a Newtonian potential generated by two point sources fixed in space. This system is integrable in the Liouville sense. We consider the Euler problem with the inverse-square potential, which can be seen as a natural generalization of the three-body problem to higher-dimensional Newtonian theory. We identify a family of stable stationary orbits in the generalized Euler problem. These orbits guarantee the existence of stable bound orbits. Applying the Poincar\'e map method to these orbits, we show that stable bound chaotic orbits appear. As a result, we conclude that the generalized Euler problem is nonintegrable.