35 Km Tabriz-Maragheh Road, Tabriz, Iran
Azarbaijan Shahid Madani University
Topological Analysis Corresponding to the Borisov–Mamaev–Sokolov Integrable System on the Lie Algebra $so(4)$
2016, vol. 21, no. 1, pp. 1-17
In 2001, A. V. Borisov, I. S. Mamaev, and V. V. Sokolov discovered a new integrable case on the Lie algebra $so(4)$. This is a Hamiltonian system with two degrees of freedom, where both the Hamiltonian and the additional integral are homogenous polynomials of degrees 2 and 4, respectively. In this paper, the topology of isoenergy surfaces for the integrable case under consideration on the Lie algebra $so(4)$ and the critical points of the Hamiltonian under consideration for different values of parameters are described and the bifurcation values of the Hamiltonian are constructed. Also, a description of bifurcation complexes and typical forms of the bifurcation diagram of the system are presented.
Akbarzadeh R., Haghighatdoost G.
The Topology of Liouville Foliation for the Borisov–Mamaev–Sokolov Integrable Case on the Lie Algebra $so(4)$
2015, vol. 20, no. 3, pp. 317-344
In 2001, A.V. Borisov, I.S. Mamaev, and V.V. Sokolov discovered a new integrable case on the Lie algebra $so(4)$. This system coincides with the Poincaré equations on the Lie algebra $so(4)$, which describe the motion of a body with cavities filled with an incompressible vortex fluid. Moreover, the Poincaré equations describe the motion of a four-dimensional gyroscope. In this paper topological properties of this system are studied. In particular, for the system under consideration the bifurcation diagrams of the momentum mapping are constructed and all Fomenko invariants are calculated. Thereby, a classification of isoenergy surfaces for this system up to the rough Liouville equivalence is obtained.