A General Solution for Rotating Lattices of Identical Point Vortices

Abstract The rotating equilibrium solutions of N identical point vortices are the minimal energy projections of a higher dimensional object to the xy plane, in this case, an N - 1 dimensional regular simplex. Parameterizing the point vortex Hamiltonian with this fundamental geometrical object leads to a simple bivector condition which determines the equilibrium states and resolves the origin of the asymmetric solutions. This novel approach uses the geometric product and the multivector derivative, ∂X, of the Hamiltonian in the geometric algebra, Cl(N - 1, 0) to find a geometric condition which determines the equilibrium states. The resulting bivector equation is then used as the input to an optimizer which rotates a simplex until the equilibrium condition is met, leading to a wealth of new solutions. If the vertices of the oriented simplex are projected to the x-axis, the values form the roots of the Hermite polynomials, HN(x), and obey the Stieltjes relations, capturing the collinear solutions. The vortex simplex exhibits a striking geometrical connection with the amplituhedron of quantum field theory and gives deep insight into the quantization of a classical system.