Abstract
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.
Highlights
Introduction and Description of the ProblemThe study of vortex dynamics can be traced back to Helmholtz’s work on hydrodynamics in 1858 [1] and it plays an important role in the study of superfluids, superconductivity, and stellar systems
We are interested to the problem in the first order Hamiltonian system of the form i zi (t) = J ∇zi H (z(t)) i ∈ 1, . . . , N
By Theorem 4.1, the linear stability of a relative equilibrium is equivalent to the fact that the central configuration generating it has a vanishing Morse index and it is non-degenerate
Summary
The study of vortex dynamics can be traced back to Helmholtz’s work on hydrodynamics in 1858 [1] and it plays an important role in the study of superfluids, superconductivity, and stellar systems. As direct consequence he proved (always under the assumption of all positive circulations) that a relative equilibrium z of the N -vortex problem is linearly stable if and only if it is generated by a nondegenerate minimum of the Hamiltonian H restricted to the level surface of the angular impulse. This result was the starting point of our analysis and the main motivation to investigate about the effect of mixed sign circulation strengths. This situation, as we’ll try to clarify, reflects somehow the difficulties and it is the paradigm of the difference between the Riemannian and the Lorentzian world
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