Constrained N-body problems

TL;DR

This paper studies gravitational N-body systems with holonomic constraints, analyzing their complex trajectories, proving non-integrability in some models, and identifying specific integrable cases using differential Galois theory.

Contribution

It introduces a framework for constrained gravitational systems, demonstrates non-integrability through differential Galois analysis, and finds particular integrable configurations.

Findings

Complex trajectories shown via Poincaré sections

Non-integrability proven for some models

Certain cases identified as integrable

Abstract

We consider a problem of mass points interacting gravitationally whose motion is subjected to certain holonomic constraints. The motion of points is restricted to certain curves and surfaces. We illustrate the complicated behaviour of trajectories of these systems using Poincar\'e cross sections. For some models we prove the non-integrability analysing properties of the differential Galois group of variational equations along certain particular solutions of considered systems. Also some integrable cases are identified.