库仑（N+1）体问题中具有柏拉图多面体对称性的辫子
Braids with the symmetries of Platonic polyhedra in the Coulomb (N+1)-body problem
摘要：
我们考虑N = 12，24，60的库仑（N + 1）体问题。其中一个粒子具有正电荷Q > 0，其余N个粒子具有相同的负电荷q < 0。这些粒子在库仑力的作用下运动，而正电荷被假定静止在质心上。施加一个对称性约束，由对称群的柏拉图多面体，我们能够计算周期轨道，使用射击方法和延续的价值Q的正电荷。在经典N体问题的背景下，通过极小化作用泛函，利用变分方法证明了这种轨道的存在性。在这里，这种方法似乎不起作用，数值计算表明，我们计算的轨道不是作用的最小化。
英文摘要：
We take into account the Coulomb (N + 1)-body problem with N = 12, 24, 60. One of the particles has positive charge Q > 0, and the remaining N have all the same negative charge q < 0. These particles move under the Coulomb force, and the positive charge is assumed to be at rest at the center of mass. Imposing a symmetry constraint, given by the symmetry group of the Platonic polyhedra, we were able to compute periodic orbits, using a shooting method and continuation with respect to the value Q of the positive charge. In the setting of the classical N -body problem, the existence of such orbits is proved with Calculus of Variation techniques, by minimizing the action functional. Here this approach does not seem to work, and numerical computations show that the orbits we compute are not minimizers of the action.
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