Untitled

Abstract

The Kowalevski top in two constant fields is known as the unique profound example of an integrable Hamiltonian system with three degrees of freedom not reducible to a family of systems in fewer dimensions. As the first approach to topological analysis of this system we find the critical set of the integral map; this set consists of the trajectories with number of frequencies less than three. We obtain the equations of the bifurcation diagram in R 3 . A correspondence to the Appelrot classes in the classical Kowalevski problem is established. The admissible regions for the values of the first integrals are found in the form of some inequalities of general character and boundary conditions for the induced diagrams on energy levels.

Separation of variables in one problem of motion of the generalized Kowalevski top

“…It is easily shown that X & O and X is the two-dimensional invariant submanifold of the flow (1) consisting of the pendulum motions found in Kharlamov (2005) a ¼ aðe 1 cos h À e 2 sin hÞ; b ¼ AEbðe 1 sin h þ e 2 cos hÞ;…”

“…It is natural to study the invariant submanifolds in P 6 such that the induced system has only two degrees of freedom. It is proved in Kharlamov (2005) …”

“…As it is shown in Kharlamov (2005), without loss of generality this system can be considered on the common level P 6 & R 9 ða; b; xÞ of the geometrical integrals The first integrals in involution are…”

“…The present work considers the restriction of the system (1) to the invariant subset O defined in P 6 by the pair of invariant relations (Kharlamov, 2005) …”

Phase topology of the Kowalevski–Sokolov top

“…Чтобы сформулировать описание критического множества отображения момента в случае Ковалевской -Соколова, напомним понятие критической подсистемы [38,20,39]. Рассмот-рим неприводимую интегрируемую систему с тремя степенями свободы с инволютивным набором интегралов, который обозначим так же, как и в нашей задаче, через H, L, K. Предположим, для простоты, что в ней отсутствуют фокусные особенности ранга 1.…”