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).   Google Scholar

[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.   Google Scholar

[3]

L. Bates and J. Sniatycki, Nonholonomic reduction,, Reports on Mathematical Physics, 32 (1993), 99.  doi: 10.1016/0034-4877(93)90073-N.  Google Scholar

[4]

I. A. Bizayev and A. V. Tsiganov, On the Routh sphere problem,, J. Phys. A: Math. Theor., 46 (2013).  doi: 10.1088/1751-8113/46/8/085202.  Google Scholar

[5]

A. V. Borisov and I. S. Mamaev, Poisson Structures and Lie Algebras in Hamiltonian Mechanics,, Izhevsk: Izd. UdSU, (1999).   Google Scholar

[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.  doi: 10.1023/A:1012995330780.  Google Scholar

[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.  doi: 10.1070/RD2002v007n02ABEH000204.  Google Scholar

[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.  doi: 10.1070/RD2002v007n02ABEH000205.  Google Scholar

[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), 7 (2002), 131.  doi: 10.1070/RD2002v007n02ABEH000199.  Google Scholar

[10]

S. A. Chaplygin, On the theory of motion of nonholonomic systems. The reducing-multiplier theorem,, Matematicheskii sbornik, 13 (2008), 369.  doi: 10.1134/S1560354708040102.  Google Scholar

[11]

R. Cushman, Routh's sphere,, Reports on Mathematical Physics, 42 (1998), 47.  doi: 10.1016/S0034-4877(98)80004-9.  Google Scholar

[12]

R. Cushman, J. J. Duistermaat and J. Sniatycki, Geometry of Nonholonomically Constrained Systems,, Advanced Series in Nonlinear Dynamics, 26 (2010).   Google Scholar

[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, 232 (2005), 75.  doi: 10.1007/0-8176-4419-9_4.  Google Scholar

[14]

F. Fassö, A. Giacobbe and N. Sansonetto, Periodic flows, rank-two Poisson structures, and nonholonomic mechanics,, Reg. Chaotic Dyn., 10 (2005), 267.  doi: 10.1070/RD2005v010n03ABEH000315.  Google Scholar

[15]

Y. N. Fedorov and B. Jovanović, Hamiltonization of the generalized Veselova LR system,, Reg. Chaotic Dyn., 14 (2009), 495.  doi: 10.1134/S1560354709040066.  Google Scholar

[16]

J. Hermans, A symmetric sphere rolling on a surface,, Nonlinearity, 8 (1995), 493.  doi: 10.1088/0951-7715/8/4/003.  Google Scholar

[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.  doi: 10.3934/jgm.2009.1.35.  Google Scholar

[18]

J. H. Jellet, A treatise on the theory of friction,, MacMillan, (1872).   Google Scholar

[19]

J. Koiller, Reduction of some classical non-holonomic systems with symmetry,, Archive for Rational Mechanics and Analysis, 118 (1992), 113.  doi: 10.1007/BF00375092.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/0021-8928(87)90060-8.  Google Scholar

[22]

V. Narayanan and P. J. Morrison, Rank change in Poisson dynamical systems,, preprint: , ().   Google Scholar

[23]

J. I. Neimark and N. A. Fufaev, Dynamics of Nonholonomic Systems,, Nauka, (1967).   Google Scholar

[24]

F. Noeter, Über rollende Bewegung einer Kugel auf Rotationsfläche,, K.B. Ludwig - Maximilians - Universitat Munchen, (1909).   Google Scholar

[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.  doi: 10.1016/j.geomphys.2011.02.015.  Google Scholar

[26]

A. Ramos, Poisson structures for reduced non-holonomic systems,, J. Phys. A: Math. Gen., 37 (2004), 4821.  doi: 10.1088/0305-4470/37/17/012.  Google Scholar

[27]

E. J. Routh, Advanced Rigid Bodies Dynamics,, MacMillan and Co., (1884).   Google Scholar

[28]

S. V. Stanchenko, Non-holonomic Chaplygin systems,, J. A Math. Mech., 53 (1989), 11.  doi: 10.1016/0021-8928(89)90126-3.  Google Scholar

[29]

A. V. Tsiganov, Compatible Lie-Poisson brackets on Lie algebras e(3) and so(4),, Teor. Math. Phys., 151 (2007), 26.  doi: 10.1007/s11232-007-0034-z.  Google Scholar

[30]

A. V. Tsiganov, On the two different bi-Hamiltonian structures for the Toda lattice,, Journal of Physics A: Math. Theor., 40 (2007), 6395.  doi: 10.1088/1751-8113/40/24/008.  Google Scholar

[31]

A. V. Tsiganov, On bi-hamiltonian geometry of the Lagrange top,, Journal of Physics A: Math. Theor., 41 (2008).  doi: 10.1088/1751-8113/41/31/315212.  Google Scholar

[32]

A. V. Tsiganov, Integrable Euler top and nonholonomic Chaplygin ball,, J. of Geometric Mechanics, 3 (2011), 337.  doi: 10.3934/jgm.2011.3.337.  Google Scholar

[33]

A. V. Tsiganov, On bi-integrable natural hamiltonian systems on Riemannian manifolds,, Journal of Nonlinear Mathematical Physics, 18 (2011), 245.  doi: 10.1142/S1402925111001507.  Google Scholar

[34]

A. V. Tsiganov, One invariant measure and different Poisson brackets for two non-holonomic systems,, Reg. Chaotic Dyn., 17 (2012), 72.  doi: 10.1134/S1560354712010078.  Google Scholar

[35]

A. V. Tsiganov, On the Poisson structures for the nonholonomic Chaplygin and Veselova problems,, Reg. Chaotic Dyn., 17 (2012), 439.  doi: 10.1134/S1560354712050061.  Google Scholar

[36]

A. V. Tsiganov, One family of conformally Hamiltonian systems,, Theor. Math. Phys., 173 (2012), 1481.  doi: 10.1007/s11232-012-0128-0.  Google Scholar

[37]

A. V. Tsiganov, On generalized nonholonomic Chaplygin sphere problem,, Int. J. Geom. Meth. in Mod. Phys., 10 (2013).  doi: 10.1142/S0219887813200089.  Google Scholar

[38]

A. V. Tsiganov, On the Lie integrability theorem for the Chaplygin ball,, Regular and Chaotic Dynamics, 19 (2014), 185.  doi: 10.1134/S1560354714020038.  Google Scholar

[39]

Ch.-J. de la Vallée-Poussin, Cours D'analyse Infinitésimale,, vol. 2, (1938).   Google Scholar

[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, 50 (1910), 101.   Google Scholar

[41]

D. V. Zenkov, The geometry of the Routh problem,, J. Nonlinear Sci., 5 (1995), 503.  doi: 10.1007/BF01209025.  Google Scholar

show all references

References:
[1]

P. Appell, Traité de Mécanique Rationnelle,, Paris: Gauthier-Villars, (1955).   Google Scholar

[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.   Google Scholar

[3]

L. Bates and J. Sniatycki, Nonholonomic reduction,, Reports on Mathematical Physics, 32 (1993), 99.  doi: 10.1016/0034-4877(93)90073-N.  Google Scholar

[4]

I. A. Bizayev and A. V. Tsiganov, On the Routh sphere problem,, J. Phys. A: Math. Theor., 46 (2013).  doi: 10.1088/1751-8113/46/8/085202.  Google Scholar

[5]

A. V. Borisov and I. S. Mamaev, Poisson Structures and Lie Algebras in Hamiltonian Mechanics,, Izhevsk: Izd. UdSU, (1999).   Google Scholar

[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.  doi: 10.1023/A:1012995330780.  Google Scholar

[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.  doi: 10.1070/RD2002v007n02ABEH000204.  Google Scholar

[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.  doi: 10.1070/RD2002v007n02ABEH000205.  Google Scholar

[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), 7 (2002), 131.  doi: 10.1070/RD2002v007n02ABEH000199.  Google Scholar

[10]

S. A. Chaplygin, On the theory of motion of nonholonomic systems. The reducing-multiplier theorem,, Matematicheskii sbornik, 13 (2008), 369.  doi: 10.1134/S1560354708040102.  Google Scholar

[11]

R. Cushman, Routh's sphere,, Reports on Mathematical Physics, 42 (1998), 47.  doi: 10.1016/S0034-4877(98)80004-9.  Google Scholar

[12]

R. Cushman, J. J. Duistermaat and J. Sniatycki, Geometry of Nonholonomically Constrained Systems,, Advanced Series in Nonlinear Dynamics, 26 (2010).   Google Scholar

[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, 232 (2005), 75.  doi: 10.1007/0-8176-4419-9_4.  Google Scholar

[14]

F. Fassö, A. Giacobbe and N. Sansonetto, Periodic flows, rank-two Poisson structures, and nonholonomic mechanics,, Reg. Chaotic Dyn., 10 (2005), 267.  doi: 10.1070/RD2005v010n03ABEH000315.  Google Scholar

[15]

Y. N. Fedorov and B. Jovanović, Hamiltonization of the generalized Veselova LR system,, Reg. Chaotic Dyn., 14 (2009), 495.  doi: 10.1134/S1560354709040066.  Google Scholar

[16]

J. Hermans, A symmetric sphere rolling on a surface,, Nonlinearity, 8 (1995), 493.  doi: 10.1088/0951-7715/8/4/003.  Google Scholar

[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.  doi: 10.3934/jgm.2009.1.35.  Google Scholar

[18]

J. H. Jellet, A treatise on the theory of friction,, MacMillan, (1872).   Google Scholar

[19]

J. Koiller, Reduction of some classical non-holonomic systems with symmetry,, Archive for Rational Mechanics and Analysis, 118 (1992), 113.  doi: 10.1007/BF00375092.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/0021-8928(87)90060-8.  Google Scholar

[22]

V. Narayanan and P. J. Morrison, Rank change in Poisson dynamical systems,, preprint: , ().   Google Scholar

[23]

J. I. Neimark and N. A. Fufaev, Dynamics of Nonholonomic Systems,, Nauka, (1967).   Google Scholar

[24]

F. Noeter, Über rollende Bewegung einer Kugel auf Rotationsfläche,, K.B. Ludwig - Maximilians - Universitat Munchen, (1909).   Google Scholar

[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.  doi: 10.1016/j.geomphys.2011.02.015.  Google Scholar

[26]

A. Ramos, Poisson structures for reduced non-holonomic systems,, J. Phys. A: Math. Gen., 37 (2004), 4821.  doi: 10.1088/0305-4470/37/17/012.  Google Scholar

[27]

E. J. Routh, Advanced Rigid Bodies Dynamics,, MacMillan and Co., (1884).   Google Scholar

[28]

S. V. Stanchenko, Non-holonomic Chaplygin systems,, J. A Math. Mech., 53 (1989), 11.  doi: 10.1016/0021-8928(89)90126-3.  Google Scholar

[29]

A. V. Tsiganov, Compatible Lie-Poisson brackets on Lie algebras e(3) and so(4),, Teor. Math. Phys., 151 (2007), 26.  doi: 10.1007/s11232-007-0034-z.  Google Scholar

[30]

A. V. Tsiganov, On the two different bi-Hamiltonian structures for the Toda lattice,, Journal of Physics A: Math. Theor., 40 (2007), 6395.  doi: 10.1088/1751-8113/40/24/008.  Google Scholar

[31]

A. V. Tsiganov, On bi-hamiltonian geometry of the Lagrange top,, Journal of Physics A: Math. Theor., 41 (2008).  doi: 10.1088/1751-8113/41/31/315212.  Google Scholar

[32]

A. V. Tsiganov, Integrable Euler top and nonholonomic Chaplygin ball,, J. of Geometric Mechanics, 3 (2011), 337.  doi: 10.3934/jgm.2011.3.337.  Google Scholar

[33]

A. V. Tsiganov, On bi-integrable natural hamiltonian systems on Riemannian manifolds,, Journal of Nonlinear Mathematical Physics, 18 (2011), 245.  doi: 10.1142/S1402925111001507.  Google Scholar

[34]

A. V. Tsiganov, One invariant measure and different Poisson brackets for two non-holonomic systems,, Reg. Chaotic Dyn., 17 (2012), 72.  doi: 10.1134/S1560354712010078.  Google Scholar

[35]

A. V. Tsiganov, On the Poisson structures for the nonholonomic Chaplygin and Veselova problems,, Reg. Chaotic Dyn., 17 (2012), 439.  doi: 10.1134/S1560354712050061.  Google Scholar

[36]

A. V. Tsiganov, One family of conformally Hamiltonian systems,, Theor. Math. Phys., 173 (2012), 1481.  doi: 10.1007/s11232-012-0128-0.  Google Scholar

[37]

A. V. Tsiganov, On generalized nonholonomic Chaplygin sphere problem,, Int. J. Geom. Meth. in Mod. Phys., 10 (2013).  doi: 10.1142/S0219887813200089.  Google Scholar

[38]

A. V. Tsiganov, On the Lie integrability theorem for the Chaplygin ball,, Regular and Chaotic Dynamics, 19 (2014), 185.  doi: 10.1134/S1560354714020038.  Google Scholar

[39]

Ch.-J. de la Vallée-Poussin, Cours D'analyse Infinitésimale,, vol. 2, (1938).   Google Scholar

[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, 50 (1910), 101.   Google Scholar

[41]

D. V. Zenkov, The geometry of the Routh problem,, J. Nonlinear Sci., 5 (1995), 503.  doi: 10.1007/BF01209025.  Google Scholar

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