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

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 too.


Introduction
Hamiltonian mechanical systems play a key role in classical mechanics, classical field theory or quantum mechanics. The equations of motion of such a system are given by a Hamiltonian function, and a perturbation of this Hamiltonian leads to a new Hamiltonian mechanical system. Another class of integrable systems that model dynamic processes is given by the Hamilton-Poisson systems, and if such a system has sufficiently constants of motion, then these give the equations of motion (see, e.g., [10]). Moreover, in [3], altering the constants of motion of the Euler top, a new integrable system, called integrable deformation of the Euler top, was obtained.
Recently, the above-mentioned idea was used to obtain integrable deformations of some three-dimensional Hamilton-Poisson systems, namely Maxwell-Bloch equations [4], Rikitake system [5], and Kermack-McKendrick system [7]. Furthermore, in [6] the integrable deformations method for a three-dimensional system of differential equations was proposed.
In this paper we give a method to construct integrable deformations of maximally superintegrable Hamiltonian mechanical systems with two degrees of freedom by alteration of their first integrals. We obtain new integrable systems which have the same number of first integrals, but which are not Hamiltonian mechanical systems, generally. Considering a particular Hamiltonian mechanical system, we show that some integrable deformations are Hamiltonian mechanical systems too. More precisely, we give integrable deformations of a Hamiltonian 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.

Integrable deformations of maximally superintegrable
Hamiltonian mechanical systems with two degrees of freedom In this section we obtain integrable deformations of maximally superintegrable Hamiltonian mechanical systems with two degrees of freedom. For details about Hamiltonian mechanical systems see, e.g. [8,9]. First, we recall some notions concerning with Hamiltonian mechanics. Let Ω ⊆ R 4 be an open set. We consider the Hamiltonian mechanical system (Ω, ω, H), where and H ∈ C ∞ (Ω, R) is the Hamiltonian function. The symplectic structure is given by the Poisson bracket on R 4 , namely for every f, g ∈ C ∞ (R 4 , R), and the equations of motion are given bẏ A function C ∈ C ∞ (Ω, R) is a first integral (constant of motion) of system (3) ifĊ = dC dt = 0, which is equivalent with the condition {H, C} ω = 0. Obviously, the Hamiltonian H is a first integral. System (3) is integrable in Liouville's sense if it possesses two functionally independent integrals of motion H, C 1 . Furthermore, system (3) is maximally superintegrable if it possesses three functionally independent first integrals H, C 1 , C 2 .
In the sequel, we consider that system (3) is maximally superintegrable. Following [10], there is a rescaling function ν ∈ C 1 (Ω, R) such that system (3) is a Hamilton-Poisson system, where and the Poisson bracket on Ω generated by C 1 , C 2 is given by for every f, g ∈ C ∞ (Ω, R), where is the Jacobian determinant.
In the following we give integrable deformations of system (4). Let α, β, γ ∈ C 1 (Ω, R) be arbitrary functions such that the functions are functionally independent on Ω, where g 1 , g 2 , g 3 ∈ R are deformation parameters. The functions I 1 , I 2 ,H give rise to the following Hamilton-Poisson systeṁ where the Poisson bracket {., .} ν I 1 ,I 2 follows from (6). Taking into account the properties of determinants, system (8) becomeṡ It is easy to see that if g 1 = g 2 = g 3 = 0, then system (8) is the initial system (4). Consequently, system (9) is a deformation of system (4). Moreover, the functions I 1 , I 2 ,H are first integrals of system (8), hence (9) is an integrable deformation, which is also a maximally superintegrable system.

Integrable deformations of a system related to the motion of two vortices in an ideal incompressible fluid
In this section we give integrable deformations of the Hamiltonian mechanical system that describes the motion of two vortices in an ideal incompressible fluid. We also consider a particular integrable deformation that preserves two of the first integrals and that is a Hamiltonian mechanical system under some conditions. In [2], following the physical description of the problem of n point vortices [1], it is mentioned that the planar motion of two vortices of coordinates (q 1 , p 1 ), (q 2 , p 2 ) in an ideal incompressible fluid can be modeled by the Hamiltonian Therefore, the motion of two vortices in an ideal incompressible fluid has a symplectic realization given by the Hamiltonian mechanical system (Ω, ω, H), where Ω is an open set in R 4 which does not contain any point (q 1 , q 1 , p 1 , p 1 ), and ω = dp 1 ∧ dq 1 + dp 2 ∧ dq 2 . Using (3), one gets the equations of motioṅ In [2], a Nambu-Hamilton realization of this system is given. We notice that it is in fact a Hamilton-Poisson realization. More precisely, recalling that the functions are first integrals of system (11), we obtain that the rescaling function is ν = 1, therefore system (11) can be put in the form (4), namelẏ where the Poisson bracket generated by C 1 , C 2 is given by Remark 1 Generally, systems (16) and (17) are not identically, hence an integrable deformation of a Hamiltonian mechanical system (Ω, ω, H), which preserves the first integrals C 1 , C 2 and is obtained by perturbationH = H + gγ, is not the Hamiltonian mechanical system (Ω, ω,H).
Imposing the condition that systems (16) and (17) to be identically, we get ∂γ Consequently, we have the following result.

Conclusions
In this paper, a method to construct integrable deformations of maximally superintegrable Hamiltonian mechanical systems with two degrees of freedom was presented. This method is used to obtain integrable deformations of the Hamiltonian mechanical system that describes the motion of two vortices in an ideal incompressible fluid. These integrable deformations are also maximally superintegrable systems, but generally they are not Hamiltonian mechanical systems. Conditions as integrable deformations of the system that describes the motion of two vortices in an ideal incompressible fluid to be Hamiltonian mechanical systems are obtained.