Abstract

We consider the planar N-centre problem of Celestial Mechanics in which we associate to each centre a homogeneous anisotropic potential. As a peculiar difference with Kepler-like potentials, the attraction field generated by every centre is no longer radial. This fact destroys the integrability of the system; however a law of conservation of the energy still holds and we prove the existence of bounded periodic trajectories for small negative energies through variational and perturbation methods. In particular, we show that, over a certain threshold on the homogeneity degree, all the solution trajectories of this system are collisionless. Moreover, as a corollary, we prove the existence of a symbolic dynamics for the system, which is made much more chaotic by the important role played by the central configurations of the potentials taken into account.

This is part of a forthcoming paper in collaboration with Vivina Barutello and Susanna Terracini.