Explicit solution of the critical subsystems for the Kovalevskaya-Chaplygin integrable family
https://doi.org/10.22405/2226-8383-2026-27-1-77-96
Abstract
In this paper we study the phase topology for the Kovalevskaya – Chaplygin integrable case in rigid body dynamics. On the one hand, it is the generalization of the classical Kovalevskaya and Chaplygin cases. On the other hand, it is inscribed in the 6-parameter family of partially
integrable (under zero value of the area integral) Hamiltonian systems with two degrees of freedom. For the given problem, we study in details the critical subsystems — Hamiltonian systems with one degree of freedom which are restrictions of the initial system to the critical
set of the momentum mapping. We obtain an explicit parametrization of the critical set which gives the bifurcation diagram and the image of the momentum mapping. For all five critical subsystems we provide their explicit solutions in elliptic quadratures under constant value of
the energy integral and the parameter of the problem. Besides that, for each critical subsystem we describe the bifurcations of the integral trajectories under the change of the energy level. It turns out that all non-trivial bifurcations of the saddle type are 2-atoms 𝐵 and 𝐶2 (standard transformations of two critical circles into one or two circles respectively).
About the Authors
Stanislav Sergeevich Nikolaenko
Moscow Institute of Physics and Technology (National Research University)
Russian Federation
Russian Federation
candidate of physical and mathematical sciences
Pavel Evgenyevich Ryabov
Moscow Institute of Physics and Technology (National Research University)
Russian Federation
Russian Federation
doctor of physical and mathematical sciences
Sergey Viktorovich Sokolov
Moscow Institute of Physics and Technology (National Research University)
Russian Federation
Russian Federation
doctor of physical and mathematical sciences
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Review
For citations:
Nikolaenko S.S., Ryabov P.E., Sokolov S.V. Explicit solution of the critical subsystems for the Kovalevskaya-Chaplygin integrable family. Chebyshevskii Sbornik. 2026;27(1):77-96. (In Russ.) https://doi.org/10.22405/2226-8383-2026-27-1-77-96
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