NDLI: Topological atlas of the Kowalevski

Content Provider	Springer Nature Link
Author	Kharlamov, Mikhail P. Ryabov, Pavel E. Savushkin, Alexander Yu.
Copyright Year	2016
Abstract	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.
Starting Page	24
Ending Page	65
Page Count	42
File Format	PDF
ISSN	15603547
Journal	Regular and Chaotic Dynamics
Volume Number	21
Issue Number	1
e-ISSN	14684845
Language	English
Publisher	Pleiades Publishing
Publisher Date	2016-02-05
Publisher Place	Moscow
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	integrable Hamiltonian systems relative equilibria isoenergy surfaces critical subsystems bifurcation diagrams rough topological invariants Dynamical Systems and Ergodic Theory
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Statistical and Nonlinear Physics Mathematical Physics Mechanical Engineering Modeling and Simulation

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