Point Vortices and Polynomials: Program Code for Testing Polynomials Associated with Point Vortices on the Plane

Abstract

   A differential equation is obtained in an explicit form for a system of 2n vortices of intensity Γ lying on a straight line and one vortex of the intensity pΓ located in the vorticity center, the polynomial solutions of which determine stationary positions of the vortices included in this configuration. The procedure of generalization of differential equations related to the stationary positions of vortices to a general differential equation for a stationary configuration of vortices is substantiated. An algorithm for generalizing differential equations is described by introducing parameters into differential equations that reflect the equivalence of the stationary positions of vortices that are obtained from each other by transformations of rotation, stretching, and shear. Generalized differential equations are presented for some stationary configurations of vortices. A procedure of testing polynomials for determining the stationary configurations of vortices is proposed. An algorithm for testing polynomials has been developed, which makes it possible to check whether the tested polynomials are solutions of the generalized differential equation of some stationary configuration of vortices. To test polynomials, based on the tested polynomials and the generalized differential equation, a system of algebraic equations is compiled with respect to the parameters of the generalized differential equation. The existence of a solution to this system corresponds to the fact that the tested polynomials satisfy the considered stationary configuration. The architecture and operation algorithm of the program that automates the testing of polynomials to determine the stationary configurations of vortices is described.