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

SIGMA 11 (2015), 085, 19 pages      arXiv:1505.01588
Contribution to the Special Issue on Analytical Mechanics and Differential Geometry in honour of Sergio Benenti

On Integrable Perturbations of Some Nonholonomic Systems

Andrey V. Tsiganov ab
a) St. Petersburg State University, St. Petersburg, Russia
b) Udmurt State University, 1 Universitetskaya Str., Izhevsk, Russia

Received May 08, 2015, in final form October 16, 2015; Published online October 20, 2015

Integrable perturbations of the nonholonomic Suslov, Veselova, Chaplygin and Heisenberg problems are discussed in the framework of the classical Bertrand-Darboux method. We study the relations between the Bertrand-Darboux type equations, well studied in the holonomic case, with their nonholonomic counterparts and apply the results to the construction of nonholonomic integrable potentials from the known potentials in the holonomic case.

Key words: nonholonomic system; integrable systems.

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  1. Bates L., Cushman R., What is a completely integrable nonholonomic dynamical system?, Rep. Math. Phys. 44 (1999), 29-35.
  2. Benenti S., Orthogonal separable dynamical systems, in Differential Geometry and its Applications (Opava, 1992), Math. Publ., Vol. 1, Editors O. Kowalsky, D. Krupka, Silesian University Opava, Opava, 1993, 163-184.
  3. Benenti S., Geometrical aspects of the dynamics of non-holonomic systems, Rend. Sem. Mat. Univ. Politec. Torino 54 (1996), 203-212.
  4. Benenti S., Intrinsic characterization of the variable separation in the Hamilton-Jacobi equation, J. Math. Phys. 38 (1997), 6578-6602.
  5. Benenti S., A 'user-friendly' approach to the dynamical equations of non-holonomic systems, SIGMA 3 (2007), 036, 33 pages, math.DS/0703043.
  6. Benenti S., A general method for writing the dynamical equations of nonholonomic systems with ideal constraints, Regul. Chaotic Dyn. 13 (2008), 283-315.
  7. Benenti S., The non-holonomic double pendulum: an example of non-linear non-holonomic system, Regul. Chaotic Dyn. 16 (2011), 417-442.
  8. Benenti S., Chanu C., Rastelli G., Variable-separation theory for the null Hamilton-Jacobi equation, J. Math. Phys. 46 (2005), 042901, 29 pages.
  9. Bertrand J.M., Mémoire sur quelques-unes des forms les plus simples que puissent présenter les intégrales des équations différentielles du mouvement d'un point matériel, J. Math. Pures Appl. 2 (1857), 113-140.
  10. Birkhoff G.D., Dynamical systems, American Mathematical Society Colloquium Publications, Vol. 9, Amer. Math. Soc., Providence, R.I., 1966.
  11. Bloch A.M., Nonholonomic mechanics and control, Interdisciplinary Applied Mathematics, Vol. 24, Springer-Verlag, New York, 2003.
  12. Bogoyavlenskiǐ O.I., Integrable cases of rigid-body dynamics and integrable systems on spheres $S^n$, Izv. Akad. Nauk SSSR Ser. Mat. 49 (1985), 899-915.
  13. Borisov A.V., Kilin A.A., Mamaev I.S., The problem of drift and recurrence for the rolling Chaplygin ball, Regul. Chaotic Dyn. 18 (2013), 832-859.
  14. Borisov A.V., Mamaev I.S., Bizyaev I.A., The hierarchy of dynamics of a rigid body rolling without slipping and spinning on a plane and a sphere, Regul. Chaotic Dyn. 18 (2013), 277-328.
  15. Borisov A.V., Mamaev I.S., Tsiganov A.V., Non-holonomic dynamics and Poisson geometry, Russ. Math. Surv. 69 (2014), 481-538.
  16. Chaplygin S.A., On a ball's rolling on a horizontal plane, Regul. Chaotic Dyn. 7 (2002), 131-148.
  17. Darboux G., Sur un probléme de mécanique, Arch. Néerl. 6 (1901), 371-376.
  18. de M. Rios P., Koiller J., Non-holonomic systems with symmetry allowing a conformally symplectic reduction, in New Advances in Celestial Mechanics and Hamiltonian Systems, Kluwer/Plenum, New York, 2004, 239-252, math-ph/0203013.
  19. Dragović V., The Appell hypergeometric functions and classical separable mechanical systems, J. Phys. A: Math. Gen. 35 (2002), 2213-2221, math-ph/0008009.
  20. Dragović V., Gajić B., Jovanović B., Generalizations of classical integrable nonholonomic rigid body systems, J. Phys. A: Math. Gen. 31 (1998), 9861-9869.
  21. Eisenhart L.P., Separable systems of Stackel, Ann. of Math. 35 (1934), 284-305.
  22. Fedorov Y.N., Jovanović B., Nonholonomic LR systems as generalized Chaplygin systems with an invariant measure and flows on homogeneous spaces, J. Nonlinear Sci. 14 (2004), 341-381, math-ph/0307016.
  23. Guha P., The role of the Jacobi last multiplier in nonholonomic systems and almost symplectic structure, Preprint, IHES/M/13/17, 2013.
  24. Jovanović B., Integrable perturbations of billiards on constant curvature surfaces, Phys. Lett. A 231 (1997), 353-358.
  25. Kalnins E.G., Miller Jr. W., Separation of variables on $n$-dimensional Riemannian manifolds. I. The $n$-sphere $S^n$ and Euclidean $n$-space ${\bf R}^n$, J. Math. Phys. 27 (1986), 1721-1736.
  26. Kozlov V.V., Methods of qualitative analysis in the dynamics of a rigid body, Moscow State University, Moscow, 1980.
  27. Kozlov V.V., On the integration theory of equations of nonholonomic mechanics, Regul. Chaotic Dyn. 7 (2002), 161-176, nlin.SI/0503027.
  28. Kozlov V.V., The Euler-Jacobi-Lie integrability theorem, Regul. Chaotic Dyn. 18 (2013), 329-343.
  29. Llibre J., Ramírez R., Sadovskaia N., Integrability of the constrained rigid body, Nonlinear Dynam. 73 (2013), 2273-2290.
  30. Molina-Becerra M., Galán-Vioque J., Freire E., Dynamics and bifurcations of a nonholonomic Heisenberg system, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 22 (2012), 1250040, 14 pages.
  31. Rosochatius E., Über die Bewegung eines Punktes, inaugural Dissertation, Univ. Göttingen, Berlin, 1877.
  32. Smirnov R.G., On the classical Bertrand-Darboux problem, J. Math. Sci. 151 (2008), 3230-3244, \mboxmath-ph/0604038.
  33. Suslov G.K., Theoretical mechanics, Gostekhizdat, Moscow, 1946.
  34. Tsiganov A., Integrable Euler top and nonholonomic Chaplygin ball, J. Geom. Mech. 3 (2011), 337-362, arXiv:1002.1123.
  35. Tsiganov A.V., On the Poisson structures for the nonholonomic Chaplygin and Veselova problems, Regul. Chaotic Dyn. 17 (2012), 439-450.
  36. Tsiganov A.V., One family of conformally Hamiltonian systems, Theoret. and Math. Phys. 173 (2012), 1481-1497, arXiv:1206.5061.
  37. Tsiganov A.V., On the Lie integrability theorem for the Chaplygin ball, Regul. Chaotic Dyn. 19 (2014), 185-197, arXiv:1312.1055.
  38. Tsiganov A.V., Killing tensors with nonvanishing Haantjes torsion and integrable systems, Regul. Chaotic Dyn. 20 (2015), 463-475.
  39. Valent G., Ben Yahia H., Neumann-like integrable models, Phys. Lett. A 360 (2007), 435-438, \mboxmath-ph/0512027.
  40. Veselova L.E., New cases of the integrability of the equations of motion of a rigid body in the presence of a nonholonomic constraint, in Geometry, Differential Equations and Mechanics (Moscow, 1985), Moscow State University, Moscow, 1986, 64-68.
  41. Wojciechowski S., Integrable one-particle potentials related to the Neumann system and the Jacobi problem of geodesic motion on an ellipsoid, Phys. Lett. A 107 (1985), 106-111.

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