Sergei Sokolov

A completely Liouville integrable Hamiltonian system with two degrees of freedom describing the dynamics of two vortex filaments in a Bose – Einstein condensate enclosed in a cylindrical trap is considered. For the system of two vortices with identical intensities a bifurcation of three Liouville tori into one is detected. Such a bifurcation is found in the integrable case of Goryachev – Chaplygin – Sretensky in rigid body dynamics.

The dynamical behavior of a heavy circular cylinder and $N$ point vortices in an unbounded volume of ideal liquid is considered. The liquid is assumed to be irrotational and at rest at infinity. The circulation about the cylinder is different from zero. The governing equations are presented in Hamiltonian form. Integrals of motion are found. Allowable types of trajectories are discussed in the case $N=1$. The stability of finding equilibrium solutions is investigated and some remarkable types of partial solutions of the system are presented. Poincar´e sections of the system demonstrate chaotic behavior of dynamics, which indicates a non-integrability of the system.

We consider a system which consists of a heavy circular cylinder and a point vortex in an unbounded volume of ideal liquid. The liquid is assumed to be irrotational and at rest at infinity. The circulation about the cylinder is different from zero. The governing equations are Hamiltonian and admit an evident integral of motion — the horizontal component of the momentum. Using the integral we reduce the order and thereby obtain a system with two degrees of freedom. Most remarkable types of partial solutions of the system are presented and stability of equilibrium solutions is investigated.