The Integrable Case of Kovalevskaya in a Non-Euclidean Spase: separation of variables

Theoretical mechanics


Sokolov S. V.

Mechanical Engineering Research Institute of the Russian Academy of Sciences, 4, M. Khariton'evskii per., Moscow, 101990, Russia



In this paper we consider the problem of the motion of a Kovalevskaya top in non-Euclidean space. Applying, as in the Euclidean case considered in the classical papers of Kovalevskaya and Ketter, non-trivial transformations of phase variables involving both generalized coordinates and conjugate momenta, we find the Abel-Jacobi equations and give the separating variables on the plane.

As is well known (see, for example, [1]), Helmholtz proposed, in the axiomatic construction of mechanics, to abandon the Euclidean property of space, and postulate only the possibility of the motion of a rigid body typical for all Riemannian spaces of constant curvature. In the light of this it is obvious that the study of the dynamics of a rigid body in spaces of constant curvature is of prime importance.

Referring to the recent review [1] for a detailed explanation of the history of research, as well as modern statements of problems in this relevant field, we note only that in classical papers attention is concentrated on obtaining equations of motion and searching for additional integrals. In this paper, following the work of Kovalevskaya [2] and Ketter [3], we obtain separated equations for the problem of the motion of the Kovalevskaya top in non-Euclidean space. As a direction for further research, one can indicate an analysis of the stability of motions that are specific to Hamiltonian systems that are completely integrable with respect to Liouville, by methods that were developed in [4,5], as well as a classical analysis of orbital stability (see, for example, [6-18]).

As a result we are reducing an integration of the original problem to hyperelliptic quadratures. An analytical expressions which were derived in presented paper can be used for the subsequent analysis of the phase topology.


integrable hamiltonian systems, separation of variables, Non-Euclidean Space


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