# American Institute of Mathematical Sciences

September  2014, 6(3): 417-440. doi: 10.3934/jgm.2014.6.417

## Poisson structures for two nonholonomic systems with partially reduced symmetries

 1 Department Computational Physics, Faculty of Physics, St.Petersburg State University, Ulyanovskaya, 3, St.Petersburg 198504, Russian Federation

Received  January 2013 Revised  May 2014 Published  September 2014

We consider nonholonomic systems which symmetry groups consist of two subgroups one of which represents rotations about the axis of symmetry. After nonholonomic reduction by another subgroup the corresponding vector fields on partially reduced phase space are linear combinations of the Hamiltonian and symmetry vector fields. The reduction of the Poisson bivectors associated with the Hamiltonian vector fields to canonical form is discussed.
Citation: Andrey Tsiganov. Poisson structures for two nonholonomic systems with partially reduced symmetries. Journal of Geometric Mechanics, 2014, 6 (3) : 417-440. doi: 10.3934/jgm.2014.6.417
##### References:
 [1] P. Appell, Traité de Mécanique Rationnelle, Paris: Gauthier-Villars, 1955. [2] P. Appell, Sur l'intégration des équations du mouvement d'un corps pesant de révolution roulant par une arëte circulaire sur un plan horizontal; Cas particulier du cerceau, Rendiconti del circolo matematico di Palermo, 14 (1900) , 1-6. [3] L. Bates and J. Sniatycki, Nonholonomic reduction, Reports on Mathematical Physics, 32 (1993), 99-115. doi: 10.1016/0034-4877(93)90073-N. [4] I. A. Bizayev and A. V. Tsiganov, On the Routh sphere problem, J. Phys. A: Math. Theor., 46 (2013), 085202 (11pp). doi: 10.1088/1751-8113/46/8/085202. [5] A. V. Borisov and I. S. Mamaev, Poisson Structures and Lie Algebras in Hamiltonian Mechanics, Izhevsk: Izd. UdSU, 1999. [6] A. V. Borisov and I. S. Mamaev, The Chaplygin problem of the rolling motion of a ball is Hamiltonian, Math. Notes, 70 (2001), 720-723. doi: 10.1023/A:1012995330780. [7] A. V. Borisov and I. S. Mamaev, The rolling motion of a rigid body on a plane and a sphere. Hierarchy of dynamics, Reg. Chaotic Dyn., 7 (2002), 177-200. doi: 10.1070/RD2002v007n02ABEH000204. [8] A. V. Borisov, I. S. Mamaev and A. A. Kilin, Rolling of a ball on a surface. New integrals and hierarchy of dynamics, Reg. Chaotic Dyn., 7 (2002), 201-219. doi: 10.1070/RD2002v007n02ABEH000205. [9] S. A. Chaplygin, On motion of heavy rigid body of revolution on horizontal plane, Trudy Otdeleniya fizicheskikh nauk Obshchestva lyubiteleii estestvoznaniya (Transactions of the Physical Section of Moscow Society of Friends of Natural Scientists), 1897, vol. 9, no. 1, English translation: Reg. Chaotic Dyn., 7 (2002), 131-148. doi: 10.1070/RD2002v007n02ABEH000199. [10] S. A. Chaplygin, On the theory of motion of nonholonomic systems. The reducing-multiplier theorem, Matematicheskii sbornik, 1911, vol. 28, issue 1, English translation: Reg. Chaotic Dyn., 13 (2008), 369-376. doi: 10.1134/S1560354708040102. [11] R. Cushman, Routh's sphere, Reports on Mathematical Physics, 42 (1998), 47-70. doi: 10.1016/S0034-4877(98)80004-9. [12] R. Cushman, J. J. Duistermaat and J. Sniatycki, Geometry of Nonholonomically Constrained Systems, Advanced Series in Nonlinear Dynamics, 26. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010. [13] K. Ehlers, J. Koiller, R. Montgomery and P. M. Rios, Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization, in The breath of Symplectic and Poisson Geometry, Progress in Mathematics, 232 (2005), 75-120. doi: 10.1007/0-8176-4419-9_4. [14] F. Fassö, A. Giacobbe and N. Sansonetto, Periodic flows, rank-two Poisson structures, and nonholonomic mechanics, Reg. Chaotic Dyn., 10 (2005), 267-284. doi: 10.1070/RD2005v010n03ABEH000315. [15] Y. N. Fedorov and B. Jovanović, Hamiltonization of the generalized Veselova LR system, Reg. Chaotic Dyn., 14 (2009), 495-505. doi: 10.1134/S1560354709040066. [16] J. Hermans, A symmetric sphere rolling on a surface, Nonlinearity, 8 (1995), 493-515. doi: 10.1088/0951-7715/8/4/003. [17] S. Hochgerner and L. García-Naranjo, G-Chaplygin systems with internal symmetries, truncation, and an (almost) symplectic view of Chaplygin's ball, J. of Geometric Mechanics, 1 (2009), 35-53. doi: 10.3934/jgm.2009.1.35. [18] J. H. Jellet, A treatise on the theory of friction, MacMillan, London, 1872. [19] J. Koiller, Reduction of some classical non-holonomic systems with symmetry, Archive for Rational Mechanics and Analysis, 118 (1992), 113-148. doi: 10.1007/BF00375092. [20] D. Korteweg, Ueber eine ziemlich verbrietete unrichtige Behandlungswiese eines Problemes der relleden Bewegung und insbesondere über kleine rollende Schwingungen um eine Gleichgewichtslage, Nieuw Archiefvoor Wiskunde, 4 (1899), 130-155. [21] N. K. Moshchuk, Reducing the equations of motion of certain nonholonomic Chaplygin systems to Lagrangian and Hamiltonian form, J. Appl. Math. Mech., 51 (1987), 172-177. doi: 10.1016/0021-8928(87)90060-8. [22] V. Narayanan and P. J. Morrison, Rank change in Poisson dynamical systems,, preprint: , (). [23] J. I. Neimark and N. A. Fufaev, Dynamics of Nonholonomic Systems, Nauka, Moscow, 1967. [24] F. Noeter, Über rollende Bewegung einer Kugel auf Rotationsfläche, K.B. Ludwig - Maximilians - Universitat Munchen, 1909. [25] T. Ohsawa, O. E. Fernandez, A. M. Bloch and D. V. Zenkov, Nonholonomic Hamilton-Jacobi theory via Chaplygin Hamiltonization, J. Geometry and Physics, 61 (2011), 1263-1291. doi: 10.1016/j.geomphys.2011.02.015. [26] A. Ramos, Poisson structures for reduced non-holonomic systems, J. Phys. A: Math. Gen., 37 (2004), 4821-4842. doi: 10.1088/0305-4470/37/17/012. [27] E. J. Routh, Advanced Rigid Bodies Dynamics, MacMillan and Co., London 1884, Reprint: Advanced Dynamics of a System of Rigid Bodies, Dover Publications, New York, 1960. [28] S. V. Stanchenko, Non-holonomic Chaplygin systems, J. A Math. Mech., 53 (1989), 11-17. doi: 10.1016/0021-8928(89)90126-3. [29] A. V. Tsiganov, Compatible Lie-Poisson brackets on Lie algebras e(3) and so(4), Teor. Math. Phys., 151 (2007), 26-43. doi: 10.1007/s11232-007-0034-z. [30] A. V. Tsiganov, On the two different bi-Hamiltonian structures for the Toda lattice, Journal of Physics A: Math. Theor., 40 (2007), 6395-6406. doi: 10.1088/1751-8113/40/24/008. [31] A. V. Tsiganov, On bi-hamiltonian geometry of the Lagrange top, Journal of Physics A: Math. Theor., 41 (2008), 315212 (12pp). doi: 10.1088/1751-8113/41/31/315212. [32] A. V. Tsiganov, Integrable Euler top and nonholonomic Chaplygin ball, J. of Geometric Mechanics, 3 (2011), 337-362. doi: 10.3934/jgm.2011.3.337. [33] A. V. Tsiganov, On bi-integrable natural hamiltonian systems on Riemannian manifolds, Journal of Nonlinear Mathematical Physics, 18 (2011), 245-268. doi: 10.1142/S1402925111001507. [34] A. V. Tsiganov, One invariant measure and different Poisson brackets for two non-holonomic systems, Reg. Chaotic Dyn., 17 (2012), 72-96. doi: 10.1134/S1560354712010078. [35] A. V. Tsiganov, On the Poisson structures for the nonholonomic Chaplygin and Veselova problems, Reg. Chaotic Dyn., 17 (2012), 439-450. doi: 10.1134/S1560354712050061. [36] A. V. Tsiganov, One family of conformally Hamiltonian systems, Theor. Math. Phys., 173 (2012), 1481-1497. doi: 10.1007/s11232-012-0128-0. [37] A. V. Tsiganov, On generalized nonholonomic Chaplygin sphere problem, Int. J. Geom. Meth. in Mod. Phys., 10 (2013), 1320008, (8 pages). doi: 10.1142/S0219887813200089. [38] A. V. Tsiganov, On the Lie integrability theorem for the Chaplygin ball, Regular and Chaotic Dynamics, 19 (2014), 185-197. doi: 10.1134/S1560354714020038. [39] Ch.-J. de la Vallée-Poussin, Cours D'analyse Infinitésimale, vol. 2, Louvain-Paris, 7th ed., 1938. [40] P. V. Voronetz, On a problem of rigid body motion rolling without sliding on the given surface under the action of given forces, Univers. Izvestiya, Proc., St. Vladimir University, 50 (1910), 101-111. [41] D. V. Zenkov, The geometry of the Routh problem, J. Nonlinear Sci., 5 (1995), 503-519. doi: 10.1007/BF01209025.

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##### References:
 [1] P. Appell, Traité de Mécanique Rationnelle, Paris: Gauthier-Villars, 1955. [2] P. Appell, Sur l'intégration des équations du mouvement d'un corps pesant de révolution roulant par une arëte circulaire sur un plan horizontal; Cas particulier du cerceau, Rendiconti del circolo matematico di Palermo, 14 (1900) , 1-6. [3] L. Bates and J. Sniatycki, Nonholonomic reduction, Reports on Mathematical Physics, 32 (1993), 99-115. doi: 10.1016/0034-4877(93)90073-N. [4] I. A. Bizayev and A. V. Tsiganov, On the Routh sphere problem, J. Phys. A: Math. Theor., 46 (2013), 085202 (11pp). doi: 10.1088/1751-8113/46/8/085202. [5] A. V. Borisov and I. S. Mamaev, Poisson Structures and Lie Algebras in Hamiltonian Mechanics, Izhevsk: Izd. UdSU, 1999. [6] A. V. Borisov and I. S. Mamaev, The Chaplygin problem of the rolling motion of a ball is Hamiltonian, Math. Notes, 70 (2001), 720-723. doi: 10.1023/A:1012995330780. [7] A. V. Borisov and I. S. Mamaev, The rolling motion of a rigid body on a plane and a sphere. Hierarchy of dynamics, Reg. Chaotic Dyn., 7 (2002), 177-200. doi: 10.1070/RD2002v007n02ABEH000204. [8] A. V. Borisov, I. S. Mamaev and A. A. Kilin, Rolling of a ball on a surface. New integrals and hierarchy of dynamics, Reg. Chaotic Dyn., 7 (2002), 201-219. doi: 10.1070/RD2002v007n02ABEH000205. [9] S. A. Chaplygin, On motion of heavy rigid body of revolution on horizontal plane, Trudy Otdeleniya fizicheskikh nauk Obshchestva lyubiteleii estestvoznaniya (Transactions of the Physical Section of Moscow Society of Friends of Natural Scientists), 1897, vol. 9, no. 1, English translation: Reg. Chaotic Dyn., 7 (2002), 131-148. doi: 10.1070/RD2002v007n02ABEH000199. [10] S. A. Chaplygin, On the theory of motion of nonholonomic systems. The reducing-multiplier theorem, Matematicheskii sbornik, 1911, vol. 28, issue 1, English translation: Reg. Chaotic Dyn., 13 (2008), 369-376. doi: 10.1134/S1560354708040102. [11] R. Cushman, Routh's sphere, Reports on Mathematical Physics, 42 (1998), 47-70. doi: 10.1016/S0034-4877(98)80004-9. [12] R. Cushman, J. J. Duistermaat and J. Sniatycki, Geometry of Nonholonomically Constrained Systems, Advanced Series in Nonlinear Dynamics, 26. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010. [13] K. Ehlers, J. Koiller, R. Montgomery and P. M. Rios, Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization, in The breath of Symplectic and Poisson Geometry, Progress in Mathematics, 232 (2005), 75-120. doi: 10.1007/0-8176-4419-9_4. [14] F. Fassö, A. Giacobbe and N. Sansonetto, Periodic flows, rank-two Poisson structures, and nonholonomic mechanics, Reg. Chaotic Dyn., 10 (2005), 267-284. doi: 10.1070/RD2005v010n03ABEH000315. [15] Y. N. Fedorov and B. Jovanović, Hamiltonization of the generalized Veselova LR system, Reg. Chaotic Dyn., 14 (2009), 495-505. doi: 10.1134/S1560354709040066. [16] J. Hermans, A symmetric sphere rolling on a surface, Nonlinearity, 8 (1995), 493-515. doi: 10.1088/0951-7715/8/4/003. [17] S. Hochgerner and L. García-Naranjo, G-Chaplygin systems with internal symmetries, truncation, and an (almost) symplectic view of Chaplygin's ball, J. of Geometric Mechanics, 1 (2009), 35-53. doi: 10.3934/jgm.2009.1.35. [18] J. H. Jellet, A treatise on the theory of friction, MacMillan, London, 1872. [19] J. Koiller, Reduction of some classical non-holonomic systems with symmetry, Archive for Rational Mechanics and Analysis, 118 (1992), 113-148. doi: 10.1007/BF00375092. [20] D. Korteweg, Ueber eine ziemlich verbrietete unrichtige Behandlungswiese eines Problemes der relleden Bewegung und insbesondere über kleine rollende Schwingungen um eine Gleichgewichtslage, Nieuw Archiefvoor Wiskunde, 4 (1899), 130-155. [21] N. K. Moshchuk, Reducing the equations of motion of certain nonholonomic Chaplygin systems to Lagrangian and Hamiltonian form, J. Appl. Math. Mech., 51 (1987), 172-177. doi: 10.1016/0021-8928(87)90060-8. [22] V. Narayanan and P. J. Morrison, Rank change in Poisson dynamical systems,, preprint: , (). [23] J. I. Neimark and N. A. Fufaev, Dynamics of Nonholonomic Systems, Nauka, Moscow, 1967. [24] F. Noeter, Über rollende Bewegung einer Kugel auf Rotationsfläche, K.B. Ludwig - Maximilians - Universitat Munchen, 1909. [25] T. Ohsawa, O. E. Fernandez, A. M. Bloch and D. V. Zenkov, Nonholonomic Hamilton-Jacobi theory via Chaplygin Hamiltonization, J. Geometry and Physics, 61 (2011), 1263-1291. doi: 10.1016/j.geomphys.2011.02.015. [26] A. Ramos, Poisson structures for reduced non-holonomic systems, J. Phys. A: Math. Gen., 37 (2004), 4821-4842. doi: 10.1088/0305-4470/37/17/012. [27] E. J. Routh, Advanced Rigid Bodies Dynamics, MacMillan and Co., London 1884, Reprint: Advanced Dynamics of a System of Rigid Bodies, Dover Publications, New York, 1960. [28] S. V. Stanchenko, Non-holonomic Chaplygin systems, J. A Math. Mech., 53 (1989), 11-17. doi: 10.1016/0021-8928(89)90126-3. [29] A. V. Tsiganov, Compatible Lie-Poisson brackets on Lie algebras e(3) and so(4), Teor. Math. Phys., 151 (2007), 26-43. doi: 10.1007/s11232-007-0034-z. [30] A. V. Tsiganov, On the two different bi-Hamiltonian structures for the Toda lattice, Journal of Physics A: Math. Theor., 40 (2007), 6395-6406. doi: 10.1088/1751-8113/40/24/008. [31] A. V. Tsiganov, On bi-hamiltonian geometry of the Lagrange top, Journal of Physics A: Math. Theor., 41 (2008), 315212 (12pp). doi: 10.1088/1751-8113/41/31/315212. [32] A. V. Tsiganov, Integrable Euler top and nonholonomic Chaplygin ball, J. of Geometric Mechanics, 3 (2011), 337-362. doi: 10.3934/jgm.2011.3.337. [33] A. V. Tsiganov, On bi-integrable natural hamiltonian systems on Riemannian manifolds, Journal of Nonlinear Mathematical Physics, 18 (2011), 245-268. doi: 10.1142/S1402925111001507. [34] A. V. Tsiganov, One invariant measure and different Poisson brackets for two non-holonomic systems, Reg. Chaotic Dyn., 17 (2012), 72-96. doi: 10.1134/S1560354712010078. [35] A. V. Tsiganov, On the Poisson structures for the nonholonomic Chaplygin and Veselova problems, Reg. Chaotic Dyn., 17 (2012), 439-450. doi: 10.1134/S1560354712050061. [36] A. V. Tsiganov, One family of conformally Hamiltonian systems, Theor. Math. Phys., 173 (2012), 1481-1497. doi: 10.1007/s11232-012-0128-0. [37] A. V. Tsiganov, On generalized nonholonomic Chaplygin sphere problem, Int. J. Geom. Meth. in Mod. Phys., 10 (2013), 1320008, (8 pages). doi: 10.1142/S0219887813200089. [38] A. V. Tsiganov, On the Lie integrability theorem for the Chaplygin ball, Regular and Chaotic Dynamics, 19 (2014), 185-197. doi: 10.1134/S1560354714020038. [39] Ch.-J. de la Vallée-Poussin, Cours D'analyse Infinitésimale, vol. 2, Louvain-Paris, 7th ed., 1938. [40] P. V. Voronetz, On a problem of rigid body motion rolling without sliding on the given surface under the action of given forces, Univers. Izvestiya, Proc., St. Vladimir University, 50 (1910), 101-111. [41] D. V. Zenkov, The geometry of the Routh problem, J. Nonlinear Sci., 5 (1995), 503-519. doi: 10.1007/BF01209025.
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