Bilinear Relative Equilibria of Identical Point Vortices

摘要

A new class of bilinear relative equilibria of identical point vortices in which the vortices are constrained to be on two perpendicular lines, conveniently taken to be the x- and y-axes of a Cartesian coordinate system, is introduced and studied. In the general problem we have m vortices on the y-axis and n on the x-axis. We define generating polynomials q(z) and p(z), respectively, for each set of vortices. A second-order, linear ODE for p(z) given q(z) is derived. Several results relating the general solution of the ODE to relative equilibrium configurations are established. Our strongest result, obtained using Sturm's comparison theorem, is that if p(z) satisfies the ODE for a given q(z) with its imaginary zeros symmetric relative to the x-axis, then it must have at least n - m + 2 simple, real zeros. For m = 2 this provides a complete characterization of all zeros, and we study this case in some detail. In particular, we show that, given q(z) = z~2 + η~2, where η is real, there is a unique p(z) of degree n, and a unique value of η~2 = A_n, such that the zeros of q(z) and p(z) form a relative equilibrium of n + 2 point vortices. We show that An ≈ 2/3n + 1/2, as n → ∞, where the coefficient of n is determined analytically, the next-order term numerically. The paper includes extensive numerical documentation on this family of relative equilibria. We dedicate this work to our co-author, friend, and colleague, Prof. Hassan Aref (1950-2011) whose work in the area of point vortex dynamics and fluid mechanics set a high standard for clarity and exposition in the field.