Analysis of constrained 2-body problem

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.