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Mat. Zametki, 2001, Volume 70, Issue 5, Pages 793–795 (Mi mz791)  

This article is cited in 56 scientific papers (total in 56 papers)

Brief Communications

Chaplygin's Ball Rolling Problem Is Hamiltonian

A. V. Borisova, I. S. Mamaevb

a M. V. Lomonosov Moscow State University
b Udmurt State University

DOI: https://doi.org/10.4213/mzm791

Full text: PDF file (163 kB)
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English version:
Mathematical Notes, 2001, 70:5, 720–723

Bibliographic databases:

Received: 17.04.2001

Citation: A. V. Borisov, I. S. Mamaev, “Chaplygin's Ball Rolling Problem Is Hamiltonian”, Mat. Zametki, 70:5 (2001), 793–795; Math. Notes, 70:5 (2001), 720–723

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. Borisov, AV, “Obstacle to the reduction of nonholonomic systems to the Hamiltonian form”, Doklady Physics, 47:12 (2002), 892  crossref  mathscinet  adsnasa  isi  scopus  scopus
    2. Mamaev IS, “New cases when the invariant measure and first integrals exist in the problem of a body rolling on a surface”, Regular & Chaotic Dynamics, 8:3 (2003), 331–335  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    3. Fedorov, YN, “Nonholonomic LR systems as generalized Chaplygin systems with an invariant measure and flows on homogeneous spaces”, Journal of Nonlinear Science, 14:4 (2004), 341  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    4. Kuang, JL, “On the chaotic instability of a nonsliding liquid-filled top with a small spheroidal base via Melnikov-Holmes-Marsden integrals”, Nonlinear Dynamics, 46:1–2 (2006), 113  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    5. A. V. Borisov, I. S. Mamaev, “Isomorphism and Hamilton representation of some nonholonomic systems”, Siberian Math. J., 48:1 (2007), 26–36  mathnet  crossref  mathscinet  zmath  isi
    6. Koiller, J, “Rubber rolling over a sphere”, Regular & Chaotic Dynamics, 12:2 (2007), 127  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    7. Garcia-Naranjo, L, “Reduction of almost Poisson brackets for nonholonomic systems on Lie groups”, Regular & Chaotic Dynamics, 12:4 (2007), 365  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    8. Borisov, AV, “Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems”, Regular & Chaotic Dynamics, 13:5 (2008), 443  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    9. Borisov, AV, “Chaplygin ball over a fixed sphere: an explicit integration”, Regular & Chaotic Dynamics, 13:6 (2008), 557  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    10. Borisov, AV, “Explicit Integration of One Problem in Nonholonomic Mechanics”, Doklady Physics, 53:10 (2008), 525  crossref  zmath  adsnasa  isi  elib  scopus  scopus
    11. Fedorov, YN, “Hamiltonization of the generalized Veselova LR system”, Regular & Chaotic Dynamics, 14:4–5 (2009), 495  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    12. Jovanovic, B, “LR and L plus R systems”, Journal of Physics A-Mathematical and Theoretical, 42:22 (2009), 225202  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    13. I. S. Mamaev, “Universalnyi kompleks programm dlya issledovaniya mekhanicheskikh sistem s negolonomnymi svyazyami”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2009, no. 2, 147–160  mathnet
    14. Hochgerner S., Garcia-Naranjo L., “G-Chaplygin Systems with Internal Symmetries, Truncation, and an (Almost) Symplectic View of Chaplygin's Ball”, J. Geom. Mech., 1:1 (2009), 35–53  crossref  mathscinet  zmath  isi
    15. A. V. Bolsinov, A. V. Borisov, I. S. Mamaev, “Gamiltonizatsiya negolonomnykh sistem v okrestnosti invariantnykh mnogoobrazii”, Nelineinaya dinam., 6:4 (2010), 829–854  mathnet  elib
    16. Hochgerner S., “Stochastic Chaplygin Systems”, Rep. Math. Phys., 66:3 (2010), 385–401  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    17. Jovanovic B., “Hamiltonization and Integrability of the Chaplygin Sphere in R-N”, J. Nonlinear Sci., 20:5 (2010), 569–593  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    18. Hochgerner S., “Chaplygin Systems Associated to Cartan Decompositions of Semi-Simple Lie Groups”, Differ. Geom. Appl., 28:4 (2010), 436–453  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    19. A. V. Tsyganov, “Deformatsii kanonicheskoi skobki Puassona dlya negolonomnykh sistem Chaplygina i Borisova–Mamaeva–Fëdorova pri nulevoi konstante ploschadei. I”, Nelineinaya dinam., 7:3 (2011), 577–599  mathnet
    20. Bolsinov A.V. Borisov A.V. Mamaev I.S., “Hamiltonization of Non-Holonomic Systems in the Neighborhood of Invariant Manifolds”, Regul. Chaotic Dyn., 16:5 (2011), 443–464  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    21. Tsiganov A., “Integrable Euler TOP and Nonholonomic Chaplygin Ball”, J. Geom. Mech., 3:3 (2011), 337–362  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    22. A. V. Tsyganov, “O puassonovykh strukturakh, voznikayuschikh pri rassmotrenii shara Chaplygina i ego obobschenii”, Nelineinaya dinam., 8:2 (2012), 345–353  mathnet
    23. A. V. Tsiganov, “One family of conformally Hamiltonian systems”, Theoret. and Math. Phys., 173:2 (2012), 1481–1497  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    24. A. V. Tsyganov, “O negolonomnykh sistemakh Veselovoi i Chaplygina”, Nelineinaya dinam., 8:3 (2012), 541–547  mathnet
    25. I. A. Bizyaev, A. V. Tsyganov, “O sfere Rausa”, Nelineinaya dinam., 8:3 (2012), 569–583  mathnet
    26. Tsiganov A.V., “On the Poisson Structures for the Nonholonomic Chaplygin and Veselova Problems”, Regul. Chaotic Dyn., 17:5 (2012), 439–450  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    27. Balseiro P., Garcia-Naranjo L.C., “Gauge Transformations, Twisted Poisson Brackets and Hamiltonization of Nonholonomic Systems”, Arch. Ration. Mech. Anal., 205:1 (2012), 267–310  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    28. Tsiganov A.V., “One Invariant Measure and Different Poisson Brackets for Two Non-Holonomic Systems”, Regul. Chaotic Dyn., 17:1 (2012), 72–96  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    29. Bizyaev I.A., Tsiganov A.V., “On the Routh Sphere Problem”, J. Phys. A-Math. Theor., 46:8 (2013), 085202  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    30. A. V. Borisov, I. S. Mamaev, I. A. Bizyaev, “Ierarkhiya dinamiki pri kachenii tverdogo tela bez proskalzyvaniya i vercheniya po ploskosti i sfere”, Nelineinaya dinam., 9:2 (2013), 141–202  mathnet
    31. A. V. Tsiganov, “On the Chaplygin Problem of the Rolling of a Ball”, Math. Notes, 94:4 (2013), 599–602  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    32. A. V. Bolsinov, A. V. Borisov, I. S. Mamaev, “Geometrizatsiya teoremy Chaplygina o privodyaschem mnozhitele”, Nelineinaya dinam., 9:4 (2013), 627–640  mathnet
    33. A. V. Tsyganov, “O share Chaplygina v absolyutnom prostranstve”, Nelineinaya dinam., 9:4 (2013), 711–719  mathnet
    34. A. V. Borisov, A. A. Kilin, I. S. Mamaev, “Problema dreifa i vozvraschaemosti pri kachenii shara Chaplygina”, Nelineinaya dinam., 9:4 (2013), 721–754  mathnet
    35. Tsiganov A.V., “On Generalized Nonholonomic Chaplygin Sphere Problem”, Int. J. Geom. Methods Mod. Phys., 10:6 (2013), 1320008  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    36. Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “The Problem of Drift and Recurrence for the Rolling Chaplygin Ball”, Regul. Chaotic Dyn., 18:6 (2013), 832–859  mathnet  crossref  mathscinet  zmath
    37. Alexey V. Borisov, Ivan S. Mamaev, Ivan A. Bizyaev, “The Hierarchy of Dynamics of a Rigid Body Rolling without Slipping and Spinning on a Plane and a Sphere”, Regul. Chaotic Dyn., 18:3 (2013), 277–328  mathnet  crossref  mathscinet  zmath
    38. A. V. Borisov, I. S. Mamaev, A. V. Tsiganov, “Non-holonomic dynamics and Poisson geometry”, Russian Math. Surveys, 69:3 (2014), 481–538  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    39. A. V. Borisov, I. S. Mamaev, “Invariantnaya mera i gamiltonizatsiya negolonomnykh sistem”, Nelineinaya dinam., 10:3 (2014), 355–359  mathnet
    40. Tsiganov A., “Poisson Structures For Two Nonholonomic Systems With Partially Reduced Symmetries”, J. Geom. Mech., 6:3 (2014), 417–440  crossref  mathscinet  zmath  isi  scopus  scopus
    41. Andrey V. Tsiganov, “On the Lie Integrability Theorem for the Chaplygin Ball”, Regul. Chaotic Dyn., 19:2 (2014), 185–197  mathnet  crossref  mathscinet  zmath
    42. Alexey V. Borisov, Ivan S. Mamaev, “Symmetries and Reduction in Nonholonomic Mechanics”, Regul. Chaotic Dyn., 20:5 (2015), 553–604  mathnet  crossref  mathscinet  zmath
    43. Balseiro P., Fernandez O.E., “Reduction of Nonholonomic Systems in Two Stages and Hamiltonization”, Nonlinearity, 28:8 (2015), 2873–2912  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    44. Bolsinov A.V., Borisov A.V., Mamaev I.S., “Geometrisation of Chaplygin'S Reducing Multiplier Theorem”, Nonlinearity, 28:7 (2015), 2307–2318  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    45. Guha P., “Nonholonomic Deformation of Coupled and Supersymmetric KdV Equations and Euler-Poincaré-Suslov Method”, Rev. Math. Phys., 27:4 (2015), 1550011  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    46. Paula Balseiro, Nicola Sansonetto, “A Geometric Characterization of Certain First Integrals for Nonholonomic Systems with Symmetries”, SIGMA, 12 (2016), 018, 14 pp.  mathnet  crossref
    47. Alexey V. Borisov, Ivan S. Mamaev, “Adiabatic Invariants, Diffusion and Acceleration in Rigid Body Dynamics”, Regul. Chaotic Dyn., 21:2 (2016), 232–248  mathnet  crossref  mathscinet  zmath  elib
    48. A. I. Zhila, “Chaplygin's ball with a rotor: Non-degeneracy of singular points”, Moscow University Mathematics Bulletin, 71:2 (2016), 45–54  mathnet  crossref  mathscinet  isi  elib
    49. Andrey V. Tsiganov, “Integrable Discretization and Deformation of the Nonholonomic Chaplygin Ball”, Regul. Chaotic Dyn., 22:4 (2017), 353–367  mathnet  crossref
    50. A. V. Borisov, I. S. Mamaev, I. A. Bizyaev, “Dynamical systems with non-integrable constraints, vakonomic mechanics, sub-Riemannian geometry, and non-holonomic mechanics”, Russian Math. Surveys, 72:5 (2017), 783–840  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    51. Balseiro P., “Hamiltonization of Solids of Revolution Through Reduction”, J. Nonlinear Sci., 27:6 (2017), 2001–2035  crossref  mathscinet  zmath  isi  scopus  scopus
    52. A. I. Zhila, “Comparison of the system “Chaplygin ball with a rotor” and the Zhukovskii system from the rough Liouville equivalence point of view”, Moscow University Mathematics Bulletin, 72:6 (2017), 245–250  mathnet  crossref  mathscinet  isi
    53. Garcia-Naranjo L.C., Montaldi J., “Gauge Momenta as Casimir Functions of Nonholonomic Systems”, Arch. Ration. Mech. Anal., 228:2 (2018), 563–602  crossref  mathscinet  zmath  isi  scopus  scopus
    54. Jovanovic B., “Rolling Balls Over Spheres in R-N”, Nonlinearity, 31:9 (2018), 4006–4030  crossref  mathscinet  zmath  isi  scopus
    55. Bizyaev I.A., Borisov A.V., Mamaev I.S., “Dynamics of the Chaplygin Ball on a Rotating Plane”, Russ. J. Math. Phys., 25:4 (2018), 423–433  crossref  mathscinet  zmath  isi  scopus
    56. Garcia-Naranjo L.C., “Generalisation of Chaplygin'S Reducing Multiplier Theorem With An Application to Multi-Dimensional Nonholonomic Dynamics”, J. Phys. A-Math. Theor., 52:20 (2019), 205203  crossref  isi
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