5 research outputs found
On bi-hamiltonian geometry of some integrable systems on the sphere with cubic integral of motion
We obtain bi-Hamiltonian structure for a family of integrable systems on the
sphere S with an additional integral of third order in momenta. These results
are applied to the Goryachev system and Goryachev-Chaplygin top for which we
give an explicit procedure to find the separated coordinates and the separated
relations.Comment: 11 pages, LaTeX with AMS fonts, corrected typo
Integrable systems on the sphere associated with genus three algebraic curves
New variables of separation for few integrable systems on the two-dimensional
sphere with higher order integrals of motion are considered in detail. We
explicitly describe canonical transformations of initial physical variables to
the variables of separation and vice versa, calculate the corresponding
quadratures and discuss some possible integrable deformations of initial
systems.Comment: 19 pages, LaTeX with AMS font