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Regular and Chaotic Dynamics

, Volume 24, Issue 2, pp 171–186 | Cite as

Hamiltonization and Separation of Variables for a Chaplygin Ball on a Rotating Plane

  • Andrey V. TsiganovEmail author
Article
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Abstract

We discuss a non-Hamiltonian vector field appearing in considering the partial motion of a Chaplygin ball rolling on a horizontal plane which rotates with constant angular velocity. In two partial cases this vector field is expressed via Hamiltonian vector fields using a nonalgebraic deformation of the canonical Poisson bivector on e*(3). For the symmetric ball we also calculate variables of separation, compatible Poisson brackets, the algebra of Haantjes operators and 2 × 2 Lax matrices.

Keywords

nonholonomic mechanics separation of variables Chaplygin ball 

MSC2010 numbers

37J60 37J35 70E18 53D17 

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References

  1. 1.
    Appell, P., Sur des transformations de movements, J. Reine Angew. Math., 1892, vol. 110, pp. 37–41.MathSciNetzbMATHGoogle Scholar
  2. 2.
    Balsero, P. and García-Naranjo, L.C., Gauge Transformations, Twisted Poisson Brackets and Hamiltonization of Nonholonomic Systems, Arch. Ration. Mech. Anal., 2012, vol. 205, no. 1, pp. 267–310.Google Scholar
  3. 3.
    Bizayev, I. A. and Tsiganov, A.V., On the Routh Sphere Problem, J. Phys. A, 2013, vol. 46, 085202, 11 pp.Google Scholar
  4. 4.
    Bizyaev, I.A., Borisov, A.V., and Mamaev, I. S., Hamiltonization of Elementary Nonholonomic Systems, Russ. J. Math. Phys., 2015, vol. 22, no. 4, pp. 444–453.MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bizyaev, I.A., Borisov, A.V., and Mamaev, I. S., The Hojman Construction and Hamiltonization of Nonholonomic Systems, SIGMA Symmetry Integrability Geom. Methods Appl., 2016, vol. 12, Paper No. 012, 19 pp.Google Scholar
  6. 6.
    Bizyaev, I.A., Borisov, A.V., and Mamaev, I. S., Dynamics of the Chaplygin Ball on a Rotating Plane, Russ. J. Math. Phys., 2018, vol. 25, no. 4, pp. 423–433.MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bloch, A.M., Fernandez, O. E., and Mestdag, T., Hamiltonization of Nonholonomic Systems and the Inverse Problem of the Calculus of Variations, Rep. Math. Phys., 2009, vol. 63, no. 2, pp. 225–249.MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., Geometrisation of Chaplygin’s Reducing Multiplier Theorem, Nonlinearity, 2015, vol. 28, no. 7, pp. 2307–2318.MathSciNetzbMATHGoogle Scholar
  9. 9.
    Borisov, A. V. and Mamaev, I. S., Chaplygin’s Ball Rolling Problem Is Hamiltonian, Math. Notes, 2001, vol. 70, no. 5, pp. 720–723; see also: Mat. Zametki, 2001, vol. 70, no. 5, pp. 793–795.MathSciNetzbMATHGoogle Scholar
  10. 10.
    Borisov, A.V., Mamaev, I. S., and Tsyganov, A.V., Nonholonomic Dynamics and Poisson Geometry, Russian Math. Surveys, 2014, vol. 69, no. 3, pp. 481–538; see also: Uspekhi Mat. Nauk, 2014, vol. 69, no. 3(417), pp. 87–144.MathSciNetGoogle Scholar
  11. 11.
    Chaplygin, S. A., On a Ball’s Rolling on a Horizontal Plane, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 131–148; see also: Math. Sb., 1903, vol. 24, no. 1, pp. 139–168.MathSciNetzbMATHGoogle Scholar
  12. 12.
    Chaplygin, S.A., On the Theory ofMotion of Nonholonomic Systems. The Reducing-Multiplier Theorem, Regul. Chaotic Dyn., 2008, vol. 13, no. 4, pp. 369–376; see also: Mat. Sb., 1912, vol. 28, no. 2, pp. 303–314.MathSciNetzbMATHGoogle Scholar
  13. 13.
    Collection of Papers Combining the Fields of Physics and Machine Learning, http://physicsml.github.io (2018).Google Scholar
  14. 14.
    Earnshaw, S., Dynamics, or An Elementary Treatise on Motion, 3rd ed., Cambridge: Deighton, 1844.Google Scholar
  15. 15.
    Ehlers, K., Koiller, J., Montgomery, R., and Rios, P.M., Nonholonomic Systems via Moving Frames: Cartan Equivalence and Chaplygin Hamiltonization, in The Breadth of Symplectic and Poisson Geometry, Progr. Math., vol. 232, Boston,Mass.: Birkhäuser, 2005, pp. 75–120.Google Scholar
  16. 16.
    Fassó, F. and Sansonetto, N., Conservation of’ moving’ energy in nonholonomic systems with affine constraints and integrability of spheres on rotating surfaces, J. Nonlinear Sci., 2016, vol. 26, no. 2, pp. 519–544.MathSciNetzbMATHGoogle Scholar
  17. 17.
    Fassò, F., García-Naranjo, L.C., and Sansonetto, N., Moving Energies As First Integrals of Nonholonomic Systems with Affine Constraints, Nonlinearity, 2018, vol. 31, no. 3, pp. 755–782.MathSciNetzbMATHGoogle Scholar
  18. 18.
    Fedorov, Yu.N. and Jovanović, B., Quasi-Chaplygin Systems and Nonholonomic Rigid Body Dynamics, Lett. Math. Phys., 2006, vol. 76, nos. 2–3, pp. 215–230.Google Scholar
  19. 19.
    Fernandez, O.E., Mestdag, T., and Bloch, A. M., A Generalization of Chaplygin’s Reducibility Theorem, Regul. Chaotic Dyn., 2009, vol. 14, no. 6, pp. 635–655.MathSciNetzbMATHGoogle Scholar
  20. 20.
    García-Naranjo, L.C. and Marrero, J.C., The Geometry of Nonholonomic Chaplygin Systems Revisited, arXiv:1812.01422 (2018).Google Scholar
  21. 21.
    Grigoryev, Yu.A. and Tsiganov, A.V., Symbolic Software for Separation of Variables in the Hamilton–Jacobi Equation for the L-Systems, Regul. Chaotic Dyn., 2005, vol. 10, no. 4, pp. 413–422.MathSciNetGoogle Scholar
  22. 22.
    Grigoryev, Yu.A. and Tsiganov, A.V., Separation of Variables for the Generalized Hénon–Heiles System and System with Quartic Potential, J. Phys. A, 2011, vol. 44, no. 25, 255202, 9 pp.Google Scholar
  23. 23.
    Grigoryev, Yu.A., Sozonov, A. P., and Tsiganov, A.V., Integrability of Nonholonomic Heisenberg Type Systems, SIGMA Symmetry Integrability Geom. Methods Appl., 2016, vol. 12, Paper No. 112, 14 pp.Google Scholar
  24. 24.
    Guha, P. and Ghose Choudhury, A., Hamiltonization of Higher-Order Nonlinear Ordinary Differential Equations and the Jacobi Last Multiplier, Acta Appl. Math., 2011, vol. 116, no. 2, pp. 179–197.MathSciNetzbMATHGoogle Scholar
  25. 25.
    Guichardet, A., Le problème de Kepler. Histoire et théorie, Paris: Éd. de l’École Polytechnique, 2012.zbMATHGoogle Scholar
  26. 26.
    Levy-Leblond, J. M., The ANAIS Billiard Table, Eur. J. Phys., 1986, vol. 7, no. 4, pp. 252–258.Google Scholar
  27. 27.
    Marle, Ch.-M., A Property of Conformally Hamiltonian Vector Fields; Application to the Kepler Problem, J. Geom. Mech., 2012, vol. 4, no. 2, pp. 181–206.MathSciNetzbMATHGoogle Scholar
  28. 28.
    Maupertuis, P. L., Accord de différentes loix de la nature qui avoient jusqu’ici paru incompatibles, in OEuvres de Maupertuis: Vol. 4, Lyon: Bruyset, 1768, pp. 3–15.Google Scholar
  29. 29.
    Ohsawa, T., Fernandez, O. E., Bloch, A.M., and Zenkov, D.V., Nonholonomic Hamilton–Jacobi Theory via Chaplygin Hamiltonization, J. Geom. Phys., 2011, vol. 61, no. 8, pp. 1263–1291.MathSciNetzbMATHGoogle Scholar
  30. 30.
    Routh, E. J., The Advanced Part of a Treatise on the Dynamics of a System of Rigid Bodies: Being Part II of a Treatise on the Whole Subject, 6th ed., New York: Dover, 1955.zbMATHGoogle Scholar
  31. 31.
    Tsiganov, A.V., Canonical Transformations of the Extended Phase Space, Toda Lattices and the Stäckel Family of Integrable Systems, J. Phys. A., 2000, vol. 33, no. 22, pp. 4169–4182.MathSciNetzbMATHGoogle Scholar
  32. 32.
    Tsiganov, A.V., The Maupertuis Principle and Canonical Transformations of the Extended Phase Space, J. Nonlinear Math. Phys., 2001, vol. 8, no. 1, pp. 157–182.MathSciNetzbMATHGoogle Scholar
  33. 33.
    Tsiganov, A.V., On Bi-Hamiltonian Structure of Some Integrable Systems on so∗(4), J. Nonlinear Math. Phys., 2008, vol. 15, no. 2, pp. 171–185.MathSciNetzbMATHGoogle Scholar
  34. 34.
    Tsiganov, A.V., On Bi-Integrable Natural Hamiltonian Systems on Riemannian Manifolds, J. Nonlinear Math. Phys., 2011, vol. 18, no. 2, pp. 245–268.MathSciNetzbMATHGoogle Scholar
  35. 35.
    Tsiganov, A.V., Integrable Euler Top and Nonholonomic Chaplygin Ball, J. Geom. Mech., 2011, vol. 3, no. 3, pp. 337–362.MathSciNetzbMATHGoogle Scholar
  36. 36.
    Tsiganov, A.V., One Family of Conformally Hamiltonian Systems, Theoret. and Math. Phys., 2012, vol. 173, no. 2, pp. 1481–1497; see also: Teoret. Mat. Fiz., 2012, vol. 173, no. 2, pp. 179–196.MathSciNetzbMATHGoogle Scholar
  37. 37.
    Tsiganov, A.V., One InvariantMeasure and Different Poisson Brackets for Two Non-Holonomic Systems, Regul. Chaotic Dyn., 2012, vol. 17, no. 1, pp. 72–96.MathSciNetzbMATHGoogle Scholar
  38. 38.
    Tsiganov, A.V., On the Poisson Structures for the Nonholonomic Chaplygin and Veselova Problems, Regul. Chaotic Dyn., 2012, vol. 17, no. 5, pp. 439–450.MathSciNetzbMATHGoogle Scholar
  39. 39.
    Tsiganov, A.V., New Variables of Separation for the Steklov–Lyapunov System, SIGMA Symmetry Integrability Geom. Methods Appl., 2012, vol. 8, Paper 012, 14 pp.Google Scholar
  40. 40.
    Tsiganov, A.V., On the Lie Integrability Theorem for the Chaplygin Ball, Regul. Chaotic Dyn., 2014, vol. 19, no. 2, pp. 185–197.MathSciNetzbMATHGoogle Scholar
  41. 41.
    Tsiganov, A.V., Poisson Structures for Two Nonholonomic Systems with Partially Reduced Symmetries, J. Geom. Mech., 2014, vol. 6, no. 3, pp. 417–440.MathSciNetzbMATHGoogle Scholar
  42. 42.
    Tsiganov, A.V., On Integrable Perturbations of Some Nonholonomic Systems, SIGMA Symmetry Integrability Geom. Methods Appl., 2015, vol. 11, Paper 085, 19 pp.Google Scholar
  43. 43.
    Tsiganov, A.V., Bäcklund Transformations for the Nonholonomic Veselova System, Regul. Chaotic Dyn., 2017, vol. 22, no. 2, pp. 163–179.MathSciNetzbMATHGoogle Scholar
  44. 44.
    Tsiganov, A.V., Integrable Discretization and Deformation of the Nonholonomic Chaplygin Ball, Regul. Chaotic Dyn., 2017, vol. 22, no. 4, pp. 353–367.MathSciNetzbMATHGoogle Scholar
  45. 45.
    Turiel, F., Structures bihamiltoniennes sur le fibré cotangent, C. R. Acad. Sci. Paris. Sér. 1. Math., 1992, vol. 315, pp. 1085–1088.MathSciNetzbMATHGoogle Scholar
  46. 46.
    Tzénoff, I., Quelques formes différentes des équations générals du mouvement des systèmes matériels, Bull. Soc. Math. France, 1925, vol. 53, pp. 80–105.MathSciNetzbMATHGoogle Scholar
  47. 47.
    Vershilov, A.V. and Tsiganov, A.V., On One Integrable System with a Cubic First Integral, Lett. Math. Phys., 2012, vol. 101, no. 2, pp. 143–156.MathSciNetzbMATHGoogle Scholar
  48. 48.
    Weierstrass, K., Über die geodätischen Linien auf dem dreiachsigen Ellipsoid, in Mathematische Werke: Vol. 1, Berlin: Mayer & Müller, 1894, pp. 257–266.Google Scholar
  49. 49.
    Zengel, K., The Electromagnetic Analogy of a Ball on a Rotating Conical Turntable, Am. J. Phys., 2017, vol. 85, no. 12, pp. 901–907.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia
  2. 2.Udmurt State UniversityIzhevskRussia

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