The formulation of the dynamics of N-bodies on the surface of an infinite cylinder is considered. For such purpose we need to make a choice of how to generalize the notion of gravitational potential on a general manifold. Following what done in Boatto, Dritschel and Schaefer, we define a gravitational potential as an attractive central force which obeys Maxwell’s like formulas.
Furthermore, when focusing on the case of two bodies’ motion, Poincar ́e sections indicate that the dynamics is non integrable. Moreover, for very low energies, when the bodies are restricted to a very small region of the cylinder, the topological signatures of the cylinder and of the plane are still present in the dynamics. A perturbative expansion is founded for the force between the two bodies. Such a force can be viewed as the planar limit plus the topological perturbation. (Joint work with G. Duarte, T. Stuchi and J. Andrade.)
