# $N$-body Dynamics on an Infinite Cylinder: the Topological Signature in the Dynamics

*2020, Volume 25, Number 1, pp. 78-110*

Author(s):

**Andrade J., Boatto S., Combot T., Duarte G., Stuchi T. J.**

The formulation of the dynamics of $N$-bodies on the surface of an infinite cylinder
is considered. We have chosen such a surface to be able to study the impact of the surface’s
topology in the particle’s dynamics. For this purpose we need to make a choice of how
to generalize the notion of gravitational potential on a general manifold. Following Boatto,
Dritschel and Schaefer [5], we define a gravitational potential as an attractive central force
which obeys Maxwell’s like formulas.

As a result of our theoretical differential Galois theory and numerical study — Poincaré sections, we prove that the two-body dynamics is not integrable. Moreover, for very low energies, when the bodies are restricted to a small region, the topological signature of the cylinder is still present in the dynamics. A perturbative expansion is derived for the force between the two bodies. Such a force can be viewed as the planar limit plus the topological perturbation. Finally, a polygonal configuration of identical masses (identical charges or identical vortices) is proved to be an unstable relative equilibrium for all $N >2$.

As a result of our theoretical differential Galois theory and numerical study — Poincaré sections, we prove that the two-body dynamics is not integrable. Moreover, for very low energies, when the bodies are restricted to a small region, the topological signature of the cylinder is still present in the dynamics. A perturbative expansion is derived for the force between the two bodies. Such a force can be viewed as the planar limit plus the topological perturbation. Finally, a polygonal configuration of identical masses (identical charges or identical vortices) is proved to be an unstable relative equilibrium for all $N >2$.

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