Gleb E. Smirnov | Searcharxiv

arXiv2013

The atlas of the diagrams for the generalization of the 4th Appelrot class of especially remarkable motions to a gyrostat in a double force field

For the system with two degrees of freedom, which is an analogue of the 4th Appelrot class for a gyrostat of the Kowalevski type in a double force field the problem of the classification of bifurcation diagrams is solved. The separating set is built and its completeness is proved. All transformations taking place in the diagrams are shown. The results serve as a necessary part of solving the problem of obtaining the topological invariants for the Reyman - Semenov-Tian-Shansky system.

nlin.SI↗

arXiv2010

The existence conditions for periodic motions of the Kowalevski gyrostat in double force field

Consider the integrable problem of motion of a gyrostat with the Kowalevski type inertia tensor in a double force field. We study the special periodic motions (the rank 1 critical points of the integral mapping) found by M.P. Kharlamov (Mekh. Tverd. Tela, No 37, 2007). Possible transformations inside the set of such motions depending on three essential parameters are studied. We obtain the analytical equations of the separating set and its singularities, point out the regions with different sets of motions. We find the image of the separating set in the space of parameters defining the bifurcation diagrams on iso-energetic levels. The most complicated calculations are fulfilled with the help of the computer system Mathematica 7.

nlin.SI↗