Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 012, 14 pages      arXiv:1101.4345      https://doi.org/10.3842/SIGMA.2012.012

New Variables of Separation for the Steklov-Lyapunov System

Andrey V. Tsiganov
St. Petersburg State University, St. Petersburg, Russia

Received October 31, 2011, in final form March 12, 2012; Published online March 20, 2012

Abstract
A rigid body in an ideal fluid is an important example of Hamiltonian systems on a dual to the semidirect product Lie algebra $e(3) = so(3)\ltimes\mathbb R^3$. We present the bi-Hamiltonian structure and the corresponding variables of separation on this phase space for the Steklov-Lyapunov system and it's gyrostatic deformation.

Key words: bi-Hamiltonian geometry; variables of separation.

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References

  1. Adler M., van Moerbeke P., Vanhaecke P., Algebraic integrability, Painlevé geometry and Lie algebras, A Series of Modern Surveys in Mathematics, Vol. 47, Springer-Verlag, Berlin, 2004.
  2. Belokolos E.D., Bobenko A.I., Enol'skii V.Z., Its A.R., Matveev V.B., Algebro-geometric approach to nonlinear integrable equations, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1994.
  3. Bobenko A.I., Euler equations on the algebras e(3) and so(4). Isomorphism of the integrable cases, Funct. Anal. Appl. 20 (1986), 53-56.
  4. Bolsinov A.V., Fedorov Y.N., Steklov-Lyapunov type systems, Preprint, 2003, available at http://upcommons.upc.edu/e-prints/bitstream/2117/900/4/0303fedorov.pdf.
  5. Borisov A.V., Tsiganov A.V. (Editors), Klebsch system. Separation of variables, explicit integration?, RCD, Moscow - Izhevsk, 2009.
  6. Bueken P., Vanhaecke P., The moduli problem for integrable systems: the example of a geodesic flow on SO(4), J. London Math. Soc. 62 (2000), 357-369.
  7. Falqui G., Pedroni M., Separation of variables for bi-Hamiltonian systems, Math. Phys. Anal. Geom. 6 (2003), 139-179, nlin.SI/0204029.
  8. Fedorov Y., Basak I., Separation of variables and explicit theta-function solution of the classical Steklov-Lyapunov systems: a geometric and algebraic geometric background, Regul. Chaotic Dyn. 16 (2011), 374-395, arXiv:0912.1788.
  9. Kirchhoff G.R., Vorlesungen über mathematische Physik Mechanik, Leipzig, 1874.
  10. Kolosoff G.V., Sur le mouvement d'un corp solide dans un liquide indéfini, C.R. Acad. Sci. Paris 169 (1919), 685-686.
  11. Kötter F., Die von Steklow und Liapunow entdeckten integralen Fälle der Bewegung eines starren Körpers in einer Flüssigkeit, Sitzungsber. König. Preuss. Akad. Wiss. 6 (1900), 79-87.
  12. Kötter F., Über die Bewegung eines festen Körpers in einer Flussigkeit, J. für Math. 109 (1892), 51-81, 89-111.
  13. Kuznetsov V., Vanhaecke P., Bäcklund transformations for finite-dimensional integrable systems: a geometric approach, J. Geom. Phys. 44 (2002), 1-40, nlin.SI/0004003.
  14. Lyapunov A.M., New integrable case of the equations of motion of a rigid body in a fluid, Fortschr. Math. 25 (1897), 1501-1504.
  15. Novikov S.P., Shmel'tser I., Periodic solutions of Kirchhoff equations for the free motion of a rigid body in a fluid and the extended Lyusternik-Shnirel'man-Morse theory (LSM). I, Funct. Anal. Appl. 15 (1981), 197-207.
  16. Rubanovsky V.N., Integrable cases in the problem of a heavy solid moving in a fluid, Dokl. Akad. Nauk SSSR 180 (1968), 556-559.
  17. Stekloff W., Ueber die Bewegung eines festen Körpers in einer Flüssigkeit, Math. Ann. 42 (1893), 273-274.
  18. Tsiganov A.V., New variables of separation for particular case of the Kowalevski top, Regul. Chaotic Dyn. 15 (2010), 659-669, arXiv:1001.4599.
  19. Tsiganov A.V., On an isomorphism of integrable cases of the Euler equations on the bi-Hamiltonian manifolds e(3) and so(4), J. Math. Sci. 136 (2006), 3641-3647.
  20. Tsiganov A.V., On bi-Hamiltonian geometry of the Lagrange top, J. Phys. A: Math. Theor. 41 (2008), 315212, 12 pages, arXiv:0802.3951.
  21. Tsiganov A.V., On bi-Hamiltonian structure of some integrable systems on so*(4), J. Nonlinear Math. Phys. 15 (2008), 171-185, nlin.SI/0703062.
  22. Tsiganov A.V., On bi-integrable natural Hamiltonian systems on Riemannian manifolds, J. Nonlinear Math. Phys. 18 (2011), 245-268, arXiv:1006.3914.
  23. Tsiganov A.V., On isomorphism of the Steklov-Lyapunov system with the potential motion on the sphere, Dokl. Math. 71 (2005), 145-147.
  24. Tsiganov A.V., On natural Poisson bivectors on the sphere, J. Phys. A: Math. Theor. 44 (2011), 105203, 21 pages, arXiv:1010.3492.
  25. Tsiganov A.V., On the Steklov-Lyapunov case of the rigid body motion, Regul. Chaotic Dyn. 9 (2004), 77-89, nlin.SI/0406017.
  26. Tsiganov A.V., On two different bi-Hamiltonian structures for the Toda lattice, J. Phys. A: Math. Gen. 40 (2007), 6395-6406, nlin.SI/0701062.
  27. Weierstrass K., Mathematische Werke I, Mayer & Muller, Berlin, 1894.

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