Dynamics and statics of vortices on a plane and a sphere - I

TL;DR

This paper explores the mathematical dynamics of point vortices on a plane and a sphere using Hamiltonian equations, Lie-Poisson, and Jacobi algebras, providing solutions and configurations.

Contribution

It introduces a unified algebraic framework for vortex dynamics on different surfaces and analyzes specific vortex configurations and solutions.

Findings

Hamiltonian equations formulated on Lie-Poisson and Jacobi algebras

Partial solutions for three and four vortex systems

Identification of stationary and static vortex configurations

Abstract

In the present paper a description of a problem of point vortices on a plane and a sphere in the "internal" variables is discussed. The hamiltonian equations of motion of vortices on a plane are built on the Lie-Poisson algebras, and in the case of vortices on a sphere on the quadratic Jacobi algebras. The last ones are obtained by deformation of the corresponding linear algebras. Some partial solutions of the systems of three and four vortices are considered. Stationary and static vortex configurations are found.