This paper concerns the investigation of the stability properties of relative equilibria which are rigidly rotating vortex configurations sometimes called vortex crystals, in the N-vortex problem. Such a configurations can be characterized as critical point of the Hamiltonian function restricted on the constant angular impulse hyper-surface in the phase space (topologically a pseudo-sphere whose coefficients are the circulation strengths of the vortices). Relative equilibria are generated by the circle action on the so-called shape pseudo-sphere (which generalize the standard shape sphere appearing in the study of the N-body problem). Inspired by the planar N-body problem, and after a geometrical and dynamical discussion of the problem, we investigate the relation intertwining the stability of relative equilibria and the inertia indices of the central configurations generating such equilibria. In the last section we applied our main results to some symmetric three and four vortices relative equilibria.

中文翻译：

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莫氏指数与平面N涡旋问题的稳定性

本文涉及N涡旋问题中相对平衡的稳定性特性的研究，这些相对平衡是刚性旋转的涡旋构型，有时也称为涡旋晶体。这样的配置可以表征为哈密顿函数的临界点限制在相空间中的恒定角度脉冲超曲面上（拓扑上是伪球体，其系数为涡旋的循环强度）。相对平衡是通过对所谓的形状伪球（它概括了研究N体问题中出现的标准形状球）的圆作用产生的。受平面N体问题的启发，并在对该问题进行几何和动力学讨论后，我们研究了相对平衡的稳定性与产生这种平衡的中心构型的惯性指数之间的关系。在最后一部分中，我们将主要结果应用于一些对称的三个和四个涡旋相对平衡。