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.

中文翻译：

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无限圆柱体上的N体动力学：动力学中的拓扑签名

氮动力学的表述-考虑了无限圆柱体表面上的物体。我们选择了这样的表面，以便能够研究表面拓扑结构对粒子动力学的影响。为此，我们需要选择如何在一般流形上概括引力势的概念。继Boatto，Dritschel和Schaefer [5]之后，我们将引力势定义为服从麦克斯韦公式的有吸引力的中心力。通过我们的理论差分伽罗瓦理论和数值研究（庞加莱截面），我们证明了两体动力学是不可整合的。此外，对于非常低的能量，当将物体限制在一个较小的区域时，圆柱体的拓扑特征仍然存在于动力学中。得出了两个物体之间力的微扰展开。这种力可以看作是平面极限加上拓扑扰动。最后，对于所有物体，相同质量（相同电荷或相同涡旋）的多边形配置被证明是不稳定的相对平衡N > 2。