On the integrable deformations of a system related to the motion of two vortices in an ideal incompressible fluid

Abstract

Altering the first integrals of an integrable system integrable deformations of the given system are obtained. These integrable deformations are also integrable systems, and they generalize the initial system. In this paper we give a method to construct integrable deformations of maximally superintegrable Hamiltonian mechanical systems with two degrees of freedom. An integrable deformation of a maximally superintegrable Hamiltonian mechanical system preserves the number of first integrals, but is not a Hamiltonian mechanical system, generally. We construct integrable deformations of the maximally superintegrable Hamiltonian mechanical system that describes the motion of two vortices in an ideal incompressible fluid, and we show that some of these integrable deformations are Hamiltonian mechanical systems tooHamiltonian mechanical system used to model the motion of two vortices in an ideal incompressible fluid, and we show that under some conditions such an integrable deformation is also a Hamiltonian mechanical system.

Stabilization of the T system by an integrable deformation

“…For the Euler top, the alteration of its constants of motion leads to some integrable deformations [7]. Moreover, integrable deformations of a three-dimensional Hamilton-Poisson system is a new Hamilton-Poisson system, which can be viewed as a controlled system [8,9,11,12,14].…”