Regular and Chaotic Dynamics

, Volume 21, Issue 1, pp 24–65 | Cite as

Topological atlas of the Kowalevski–Sokolov top

  • Mikhail P. KharlamovEmail author
  • Pavel E. Ryabov
  • Alexander Yu. Savushkin


We investigate the phase topology of the integrable Hamiltonian system on e(3) found by V. V. Sokolov (2001) and generalizing the Kowalevski case. This generalization contains, along with a homogeneous potential force field, gyroscopic forces depending on the configurational variables. The relative equilibria are classified, their type is calculated and the character of stability is defined. The Smale diagrams of the case are found and the isoenergy manifolds of the reduced systems with two degrees of freedom are classified. The set of critical points of the momentum map is represented as a union of critical subsystems; each critical subsystem is a one-parameter family of almost Hamiltonian systems with one degree of freedom. For all critical points we explicitly calculate the characteristic values defining their type. We obtain the equations of the diagram of the momentum map and give a classification of isoenergy and isomomentum diagrams equipped with the description of regular integral manifolds and their bifurcations. We construct the Smale–Fomenko diagrams which, when considered in the enhanced space of the energy-momentum constants and the essential physical parameters, separate 25 different types of topological invariants called the Fomenko graphs. We find all marked loop molecules of rank 0 nondegenerate critical points and of rank 1 degenerate periodic trajectories. Analyzing the cross-sections of the isointegral equipped diagrams, we get a complete list of the Fomenko graphs. The marks on them producing the exact topological invariants of Fomenko–Zieschang can be found from previous investigations of two partial cases with some additions obtained from the loop molecules or by a straightforward calculation using the separation of variables.


integrable Hamiltonian systems relative equilibria isoenergy surfaces critical subsystems bifurcation diagrams rough topological invariants 

MSC2010 numbers

70E05 70E17 37J35 34A05 


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Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  • Mikhail P. Kharlamov
    • 1
    Email author
  • Pavel E. Ryabov
    • 2
  • Alexander Yu. Savushkin
    • 1
  1. 1.Russian Academy of National Economy and Public AdministrationVolgogradRussia
  2. 2.Financial University under the Government of Russian FederationMoscowRussia

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