Analysis of constrained 2-body problem

TL;DR

This paper investigates the dynamics of two elastic particles constrained to planar curves, demonstrating nonintegrability, analyzing stability, and identifying periodic solutions using Poincaré sections and Birkhoff normal form.

Contribution

It provides a proof of nonintegrability and offers a detailed analysis of stability and periodic solutions in a constrained two-body elastic system.

Findings

System is nonintegrable.

Conditions for linear stability are established.

Particular periodic solutions are identified.

Abstract

We consider the system of two material points that interact by elastic forces according to Hooke's law and their motion is restricted to certain curves lying on the plane. The nonintegrability of this system and idea of the proof are communicated. Moreover, the analysis of global dynamics by means of Poincar\'e cross sections is given and local analysis in the neighborhood of an equilibrium is performed by applying the Birkhoff normal form. Conditions of linear stability are determined and some particular periodic solutions are identified.