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

Theoretical mechanics


Аuthors

Sokolov S. V.

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

e-mail: sokolovsv72@mail.ru

Abstract

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.

Keywords:

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

References

  1. Borisov A.V., Mamaev I.S. Rigid Body Dynamics in Non-Euclidean Spaces, Russian Journal of Mathematical Physics, 2016, vol. 23, no. 4, pp. 431 – 454.

  2. Kowalevski S. Sur le probléme de la rotation d’un corps solide autour d’un point fixe, Acta Mathematica, 1889, vol. 12, pp. 177 – 232.

  3. Kötter F. Sur le cas trait’e par M-me Kowalevski de rotation d’un corps solide autour d’un point fixe, Acta Mathematica, 1893, vol. 17, no. 1–2, pp. 209 – 263.

  4. Smale S. Topology and mechanics, Nventiones Mathematicae, 1970, vol. 10, no. 4, pp. 305 – 331.

  5. Bolsinov A.V., Borisov A.V., Mamaev I.S. Uspekhi matematicheskikh nauk, 2010, vol. 65, no. 2 (392), pp. 71 – 132.

  6. Bardin B.S., Savin A.A. On the orbital stability of pendulum-like oscillations and rotations of a symmetric rigid body with a fixed point, Regular and Chaotic Dynamics, 2012, vol. 17, no. 3–4, pp. 243 – 257.

  7. Bardin B.S., Chekina E.A. On the Stability of Resonant Rotation of a Symmetric Satellite in an Elliptical Orbit, Regular and Chaotic Dynamics, 2016, vol. 21, no. 4, pp. 377 – 389.

  8. Bardin B.S., Lanchares V. On the Stability of Periodic Hamiltonian Systems with One Degree of Freedom in the Case of Degeneracy, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 627 – 648.

  9. Bardin B.S., Chekina E.A. Trudy MAI, 2016, no. 89, available at: http://trudymai.ru/eng/published.php?ID=72568

  10. Bardin B.S., Chekina E.A. On the Constructive Algorithm for Stability Analysis of an Equilibriмum Point of a Periodic Hamiltonian System with Two Degrees of Freedom in the Second-order Resonance Case, Regular and Chaotic Dynamics, 2017, vol. 22, no. 7, pp. 808 – 823.

  11. Markeev A.P. Prikladnaya matematika i mekhanika, 2001, vol. 65, no. 1, pp. 51 – 58.

  12. Markeev A.P. On stability of regular precessions of a non-symmetric gyroscope, Regular and Chaotic Dynamics, 2003, vol. 8, no. 3, pp. 297 – 304.

  13. Bardin B.S., Chekina E.A., Chekin A.M. On the Stability of a Planar Resonant Rotation of a Satellite in an Elliptic Orbit, Regular and Chaotic Dynamics, 2015, vol. 20, no. 1, pp. 63 – 73.

  14. Bardin B.S., Rudenko T.V., Savin A.A. On the Orbital Stability of Planar Periodic Motions of a Rigid Body in the Bobylev–Steklov Case, Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 533 – 546.

  15. Bardin B.S. On the orbital stability of pendulum-like motions of a rigid body in the Bobylev–Steklov case, Regular and Chaotic Dynamics, 2010, vol. 15, no. 6, pp. 704 – 716.

  16. Markeev A.P. On the Stability of Periodic Motions of an Autonomous Hamiltonian System in a Critical Case of the Fourth-order Resonance, Regular and Chaotic Dynamics, 2017, vol. 22, no. 7, pp. 773 – 781.

  17. Bardin B.S. On Nonlinear Motions of Hamiltonian System in Case of Fourth Order Resonance, Regular and Chaotic Dynamics, 2007, vol. 12, no. 1, pp. 86 – 100.

  18. Bardin B.S., Maciejewski A.J., Przybylska M. Integrability of generalized Jacobi problem, Regular and Chaotic Dynamics, 2005, vol. 10, no. 4, pp. 437 – 461.

  19. Borisov A.V., Mamaev I.S. Dinamika tverdogo tela (Rigid Body Dynamics), Izhevsk, NITs “Regulyarnaya i khaoticheskaya dinamika”, 2001, 384 p.

  20. Suslov G.K. Teoreticheskaya mekhanika (Theoretical Mechanics), Moscow, Gostekhizdat, 1946, 655 p.


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