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Existence, stability, and symmetry of relative equilibria with a dominant vortex. (English) Zbl 1396.70021

Summary: We analyze existence, stability, and symmetry of point vortex relative equilibria with one dominant vortex and \(N\) vortices with infinitesimal circulation. The dimension of the problem can be reduced by taking an infinitesimal circulation limit, resulting in the so-called \((1+N)\)-vortex problem. In this work, we first generalize the reduction to allow for circulations of varying signs and weights. We then prove that symmetric configurations require equality of two circulation parameters in the \((1+3)\)-vortex problem and show that there are stable asymmetric relative equilibria. In a number of examples, we use rigorous methods from algebraic geometry to count all relative equilibria.

MSC:

70H14 Stability problems for problems in Hamiltonian and Lagrangian mechanics
70F15 Celestial mechanics
70H12 Periodic and almost periodic solutions for problems in Hamiltonian and Lagrangian mechanics
76B47 Vortex flows for incompressible inviscid fluids
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
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