Universität Augsburg
Institut für Mathematik

Please ask Prof. Kai Cieliebak for­ login details.

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 and the spherical symmetry disappears. This fact destroys the integrability of the system; however, a law of conservation of the energy still holds and we prove the existence of 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 collision-less. Moreover, we prove the existence of a symbolic dynamics for the system, which is made much more complicated 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.