2011, 3(3): 337-362. doi: 10.3934/jgm.2011.3.337

Integrable Euler top and nonholonomic Chaplygin ball

1. 

St. Petersburg State University, St. Petersburg, Russian Federation

Received  April 2011 Revised  July 2011 Published  November 2011

We discuss the Poisson structures, Lax matrices, $r$-matrices, bi-hamiltonian structures, the variables of separation and other attributes of the modern theory of dynamical systems in application to the integrable Euler top and to the nonholonomic Chaplygin ball.
Citation: Andrey Tsiganov. Integrable Euler top and nonholonomic Chaplygin ball. Journal of Geometric Mechanics, 2011, 3 (3) : 337-362. doi: 10.3934/jgm.2011.3.337
References:
[1]

R. Abraham and J. E. Marsden, "Foundations of Mechanics," Second edition, revised and enlarged,, With the assistance of Tudor Raţiu and Richard Cushman, (1978).

[2]

M. Audin, "Spinning Tops. A Course on Integrable Systems,", Cambridge Studies in Advanced Mathematics, 51 (1996).

[3]

O. Babelon and C.-M. Viallet, Hamiltonian structures and Lax equations,, Phys. Lett. B, 237 (1990), 411. doi: 10.1016/0370-2693(90)91198-K.

[4]

O. I. Bogoyavlenskiĭ, Integrable cases of rigid-body dynamics and integrable systems on the spheres $S^n$,, Izv. Akad. Nauk SSSR Ser. Mat., 49 (1985), 899.

[5]

A. V. Bolsinov and B. Jovanović, Noncommutative integrability, moment map and geodesic flows,, Ann. Glob. Anal. and Geom., 23 (2003), 305. doi: 10.1023/A:1023023300665.

[6]

A. V. Borisov and I. S. Mamaev, Chaplygin's ball rolling problem is Hamiltonian,, Math. Notes, 70 (2001), 720. doi: 10.1023/A:1012995330780.

[7]

A. V. Borisov and I. S. Mamaev, "Dynamics of a Rigid Body. Hamiltonian Methods, Integrability, Chaos,", Second edition, (2005).

[8]

A. V. Borisov and I. S. Mamaev, Conservation laws, hierarchy of dynamics and explicit integration of nonholonomic systems,, Reg. Chaotic Dyn., 13 (2008), 443. doi: 10.1134/S1560354708050079.

[9]

A. V. Borisov, Yu. N. Fedorov and I. S. Mamaev, Chaplygin ball over a fixed sphere: An explicit integration,, Reg. Chaotic Dyn., 13 (2008), 557. doi: 10.1134/S1560354708060063.

[10]

S. A. Chaplygin, "On a Motion of a Heavy Body of Revolution on a Horizontal Plane,", Translated from, 7 (2002), 51. doi: 10.1070/RD2002v007n02ABEH000199.

[11]

S. A. Chaplygin, On a ball's rolling on a horizontal plane,, Regul. Chaotic Dyn., 7 (2002), 131. doi: 10.1070/RD2002v007n02ABEH000200.

[12]

J. J. Duistermaat, Chaplygin’s sphere,, preprint, ().

[13]

K. Ehlers, J. Koiller, R. Montgomery and P. M. Rios, Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization,, in, 232 (2005), 75.

[14]

J. C. Eilbeck, V. Z. Énol'skiĭ , V. B. Kuznetsov and A. V. Tsiganov, Linear r-matrix algebra for classical separable systems,, J. Phys. A, 27 (1994), 567. doi: 10.1088/0305-4470/27/2/038.

[15]

G. Falqui and M. Pedroni, Separation of variables for bi-Hamiltonian systems,, Math. Phys. Anal. Geom., 6 (2003), 139. doi: 10.1023/A:1024080315471.

[16]

F. Fassò, The Euler-Poinsot top: A non-commutatively integrable system without global action-angle coordinates,, Zeitschrift für Angewandte Mathematik und Physik, 47 (1996), 953. doi: 10.1007/BF00920045.

[17]

Yu. N. Fedorov, Integration of a generalized problem on the rolling of a Chaplygin ball,, in, (1986), 151.

[18]

C. G. J. Jacobi, Vorlesungen über Dynamik,, in, (1866), 1842.

[19]

B. Jovanovic, Hamiltonization and integrability of the Chaplygin sphere in $R^n$,, J. of Nonlinear Science, 20 (2010), 569.

[20]

E. G. Gallop, On the rise of a spinning top,, Trans. Cambridge Phil. Society, 19 (1904), 356.

[21]

S. Hochgerner, Chaplygin systems associated to Cartan decompositions of semi-simple Lie groups,, Diff. Geom. Appl., 28 (2010), 436. doi: 10.1016/j.difgeo.2010.04.003.

[22]

E. G. Kalnins, "Separation of Variables for Riemannian Spaces of Constant Curvature,", Pitman Monographs and Surveys in Pure and Applied Mathematics, 28 (1986).

[23]

I. V. Komarov and A. V.Tsiganov, On a trajectory isomorphism of the Kowalevski gyrostat and the Clebsch problem,, Journal of Physics A, 38 (2005), 2917. doi: 10.1088/0305-4470/38/13/007.

[24]

J.-L. Koszul, Crochet de Schouten-Nijenhuis et cohomologie,, Astérisque, 1985 (): 257.

[25]

V. V. Kozlov, Realization of nonintegrable constraints in classical mechanics,, Dokl. Akad. Nauk SSSR, 272 (1983), 550.

[26]

V. V. Kozlov, On the integration theory of equations of nonholonomic mechanics,, Adv. in Mech., 8 (1985), 85.

[27]

T. E. Kouloukas and V. G. Papageorgiou, Poisson Yang-Baxter maps with binomial Lax matrices,, J. Math. Phys., 52 (2011).

[28]

V. B. Kuznetsov, Quadrics on real Riemannian spaces of constant curvature: Separation of variables and connection with Gaudin magnet,, J. Math. Phys., 33 (1992), 3240. doi: 10.1063/1.529542.

[29]

A. Lichnerowicz, Les variétés de Poisson et leurs algèbres de Lie associées,, J. Diff. Geom., 12 (1977), 253.

[30]

W. Macke, "Mechanik der Teilchen, Systeme und Kontinua: Ein Lehrbuch der theoretischen Physik,", Akademische Verlagsgesellschaft Geest & Portig K.-G., (1962).

[31]

A. P. Markeev, Integrability of a problem on rolling of ball with multiply connected cavity filled by ideal liquid,, Izv. Akad. Nauk SSSR, 21 (1986), 64.

[32]

C. Morosi and L. Pizzocchero, On the Euler equation: Bi-Hamiltonian structure and integrals in involution,, Lett. Math. Phys., 37 (1996), 117. doi: 10.1007/BF00416015.

[33]

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.

[34]

A. G. Reyman and M. A. Semenov-Tian-Shansky, Group-theoretical methods in the theory of finite-dimensional integrable systems,, in, 7 (1987).

[35]

D. Schneider, Non-holonomic Euler-Poincaré equations and stability in Chaplygin's sphere,, Dyn. Syst., 17 (2002), 87. doi: 10.1080/02681110110112852.

[36]

J. L. Synge, "Classical Dynamics,", 1960 Handbuch der Physik, (1960), 1.

[37]

A. V. Tsiganov, The Stäckel systems and algebraic curves,, J. Math. Phys., 40 (1999), 279. doi: 10.1063/1.532789.

[38]

A. V. Tsiganov, Duality between integrable Stäckel systems,, J. Phys. A, 32 (1999), 7965. doi: 10.1088/0305-4470/32/45/311.

[39]

A. V. Tsiganov, The Maupertuis principle and canonical transformations of the extended phase space,, J. Nonlinear Math. Phys., 8 (2001), 157. doi: 10.2991/jnmp.2001.8.1.12.

[40]

A. V. Tsiganov, On the Steklov-Lyapunov case of the rigid body motion,, Regular and Chaotic Dynamics, 9 (2004), 77. doi: 10.1070/RD2004v009n02ABEH000267.

[41]

A. V. Tsiganov, Toda chains in the Jacobi method,, Teor. Math. Phys., 139 (2004), 636. doi: 10.1023/B:TAMP.0000026181.79622.af.

[42]

A. V. Tsiganov, A family of the Poisson brackets compatible with the Sklyanin bracket,, J. Phys. A, 40 (2007), 4803. doi: 10.1088/1751-8113/40/18/008.

[43]

A. V. Tsiganov, On bi-hamiltonian geometry of the Lagrange top,, J. Phys. A, 41 (2008).

[44]

A. V. Tsiganov, New variables of separation for particular case of the Kowalevski top,, Regular and Chaotic Dynamics, 15 (2010), 659. doi: 10.1134/S156035471006002X.

[45]

A. V. Tsiganov, On natural Poisson bivectors on the sphere,, J. Phys. A, 44 (2011).

[46]

A. V. Tsiganov, On deformations of the canonical Poisson bracket for the nonholonomic Chaplygin and the Borisov–Mamaev–Fedorov systems on zero-level of the area integral I,, Rus. J. Nonlin. Dynamics, 7 (2011), 577.

[47]

I. Vaisman, "Lectures on the Geometry of Poisson Manifolds,", Progress in Mathematics, 118 (1994).

[48]

A. Weinstein, The modular automorphism group of a Poisson manifold,, J. Geom. Phys., 23 (1997), 379. doi: 10.1016/S0393-0440(97)80011-3.

[49]

S. Wojciechowski, Integrable one-particle potentials related to the Neumann systems and the Jacobi problem of geodesic motion on an ellipsoid,, Phys. Lett. A, 107 (1985), 106. doi: 10.1016/0375-9601(85)90725-X.

show all references

References:
[1]

R. Abraham and J. E. Marsden, "Foundations of Mechanics," Second edition, revised and enlarged,, With the assistance of Tudor Raţiu and Richard Cushman, (1978).

[2]

M. Audin, "Spinning Tops. A Course on Integrable Systems,", Cambridge Studies in Advanced Mathematics, 51 (1996).

[3]

O. Babelon and C.-M. Viallet, Hamiltonian structures and Lax equations,, Phys. Lett. B, 237 (1990), 411. doi: 10.1016/0370-2693(90)91198-K.

[4]

O. I. Bogoyavlenskiĭ, Integrable cases of rigid-body dynamics and integrable systems on the spheres $S^n$,, Izv. Akad. Nauk SSSR Ser. Mat., 49 (1985), 899.

[5]

A. V. Bolsinov and B. Jovanović, Noncommutative integrability, moment map and geodesic flows,, Ann. Glob. Anal. and Geom., 23 (2003), 305. doi: 10.1023/A:1023023300665.

[6]

A. V. Borisov and I. S. Mamaev, Chaplygin's ball rolling problem is Hamiltonian,, Math. Notes, 70 (2001), 720. doi: 10.1023/A:1012995330780.

[7]

A. V. Borisov and I. S. Mamaev, "Dynamics of a Rigid Body. Hamiltonian Methods, Integrability, Chaos,", Second edition, (2005).

[8]

A. V. Borisov and I. S. Mamaev, Conservation laws, hierarchy of dynamics and explicit integration of nonholonomic systems,, Reg. Chaotic Dyn., 13 (2008), 443. doi: 10.1134/S1560354708050079.

[9]

A. V. Borisov, Yu. N. Fedorov and I. S. Mamaev, Chaplygin ball over a fixed sphere: An explicit integration,, Reg. Chaotic Dyn., 13 (2008), 557. doi: 10.1134/S1560354708060063.

[10]

S. A. Chaplygin, "On a Motion of a Heavy Body of Revolution on a Horizontal Plane,", Translated from, 7 (2002), 51. doi: 10.1070/RD2002v007n02ABEH000199.

[11]

S. A. Chaplygin, On a ball's rolling on a horizontal plane,, Regul. Chaotic Dyn., 7 (2002), 131. doi: 10.1070/RD2002v007n02ABEH000200.

[12]

J. J. Duistermaat, Chaplygin’s sphere,, preprint, ().

[13]

K. Ehlers, J. Koiller, R. Montgomery and P. M. Rios, Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization,, in, 232 (2005), 75.

[14]

J. C. Eilbeck, V. Z. Énol'skiĭ , V. B. Kuznetsov and A. V. Tsiganov, Linear r-matrix algebra for classical separable systems,, J. Phys. A, 27 (1994), 567. doi: 10.1088/0305-4470/27/2/038.

[15]

G. Falqui and M. Pedroni, Separation of variables for bi-Hamiltonian systems,, Math. Phys. Anal. Geom., 6 (2003), 139. doi: 10.1023/A:1024080315471.

[16]

F. Fassò, The Euler-Poinsot top: A non-commutatively integrable system without global action-angle coordinates,, Zeitschrift für Angewandte Mathematik und Physik, 47 (1996), 953. doi: 10.1007/BF00920045.

[17]

Yu. N. Fedorov, Integration of a generalized problem on the rolling of a Chaplygin ball,, in, (1986), 151.

[18]

C. G. J. Jacobi, Vorlesungen über Dynamik,, in, (1866), 1842.

[19]

B. Jovanovic, Hamiltonization and integrability of the Chaplygin sphere in $R^n$,, J. of Nonlinear Science, 20 (2010), 569.

[20]

E. G. Gallop, On the rise of a spinning top,, Trans. Cambridge Phil. Society, 19 (1904), 356.

[21]

S. Hochgerner, Chaplygin systems associated to Cartan decompositions of semi-simple Lie groups,, Diff. Geom. Appl., 28 (2010), 436. doi: 10.1016/j.difgeo.2010.04.003.

[22]

E. G. Kalnins, "Separation of Variables for Riemannian Spaces of Constant Curvature,", Pitman Monographs and Surveys in Pure and Applied Mathematics, 28 (1986).

[23]

I. V. Komarov and A. V.Tsiganov, On a trajectory isomorphism of the Kowalevski gyrostat and the Clebsch problem,, Journal of Physics A, 38 (2005), 2917. doi: 10.1088/0305-4470/38/13/007.

[24]

J.-L. Koszul, Crochet de Schouten-Nijenhuis et cohomologie,, Astérisque, 1985 (): 257.

[25]

V. V. Kozlov, Realization of nonintegrable constraints in classical mechanics,, Dokl. Akad. Nauk SSSR, 272 (1983), 550.

[26]

V. V. Kozlov, On the integration theory of equations of nonholonomic mechanics,, Adv. in Mech., 8 (1985), 85.

[27]

T. E. Kouloukas and V. G. Papageorgiou, Poisson Yang-Baxter maps with binomial Lax matrices,, J. Math. Phys., 52 (2011).

[28]

V. B. Kuznetsov, Quadrics on real Riemannian spaces of constant curvature: Separation of variables and connection with Gaudin magnet,, J. Math. Phys., 33 (1992), 3240. doi: 10.1063/1.529542.

[29]

A. Lichnerowicz, Les variétés de Poisson et leurs algèbres de Lie associées,, J. Diff. Geom., 12 (1977), 253.

[30]

W. Macke, "Mechanik der Teilchen, Systeme und Kontinua: Ein Lehrbuch der theoretischen Physik,", Akademische Verlagsgesellschaft Geest & Portig K.-G., (1962).

[31]

A. P. Markeev, Integrability of a problem on rolling of ball with multiply connected cavity filled by ideal liquid,, Izv. Akad. Nauk SSSR, 21 (1986), 64.

[32]

C. Morosi and L. Pizzocchero, On the Euler equation: Bi-Hamiltonian structure and integrals in involution,, Lett. Math. Phys., 37 (1996), 117. doi: 10.1007/BF00416015.

[33]

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.

[34]

A. G. Reyman and M. A. Semenov-Tian-Shansky, Group-theoretical methods in the theory of finite-dimensional integrable systems,, in, 7 (1987).

[35]

D. Schneider, Non-holonomic Euler-Poincaré equations and stability in Chaplygin's sphere,, Dyn. Syst., 17 (2002), 87. doi: 10.1080/02681110110112852.

[36]

J. L. Synge, "Classical Dynamics,", 1960 Handbuch der Physik, (1960), 1.

[37]

A. V. Tsiganov, The Stäckel systems and algebraic curves,, J. Math. Phys., 40 (1999), 279. doi: 10.1063/1.532789.

[38]

A. V. Tsiganov, Duality between integrable Stäckel systems,, J. Phys. A, 32 (1999), 7965. doi: 10.1088/0305-4470/32/45/311.

[39]

A. V. Tsiganov, The Maupertuis principle and canonical transformations of the extended phase space,, J. Nonlinear Math. Phys., 8 (2001), 157. doi: 10.2991/jnmp.2001.8.1.12.

[40]

A. V. Tsiganov, On the Steklov-Lyapunov case of the rigid body motion,, Regular and Chaotic Dynamics, 9 (2004), 77. doi: 10.1070/RD2004v009n02ABEH000267.

[41]

A. V. Tsiganov, Toda chains in the Jacobi method,, Teor. Math. Phys., 139 (2004), 636. doi: 10.1023/B:TAMP.0000026181.79622.af.

[42]

A. V. Tsiganov, A family of the Poisson brackets compatible with the Sklyanin bracket,, J. Phys. A, 40 (2007), 4803. doi: 10.1088/1751-8113/40/18/008.

[43]

A. V. Tsiganov, On bi-hamiltonian geometry of the Lagrange top,, J. Phys. A, 41 (2008).

[44]

A. V. Tsiganov, New variables of separation for particular case of the Kowalevski top,, Regular and Chaotic Dynamics, 15 (2010), 659. doi: 10.1134/S156035471006002X.

[45]

A. V. Tsiganov, On natural Poisson bivectors on the sphere,, J. Phys. A, 44 (2011).

[46]

A. V. Tsiganov, On deformations of the canonical Poisson bracket for the nonholonomic Chaplygin and the Borisov–Mamaev–Fedorov systems on zero-level of the area integral I,, Rus. J. Nonlin. Dynamics, 7 (2011), 577.

[47]

I. Vaisman, "Lectures on the Geometry of Poisson Manifolds,", Progress in Mathematics, 118 (1994).

[48]

A. Weinstein, The modular automorphism group of a Poisson manifold,, J. Geom. Phys., 23 (1997), 379. doi: 10.1016/S0393-0440(97)80011-3.

[49]

S. Wojciechowski, Integrable one-particle potentials related to the Neumann systems and the Jacobi problem of geodesic motion on an ellipsoid,, Phys. Lett. A, 107 (1985), 106. doi: 10.1016/0375-9601(85)90725-X.

[1]

Y. A. Li, P. J. Olver. Convergence of solitary-wave solutions in a perturbed bi-Hamiltonian dynamical system I. Compactions and peakons. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 419-432. doi: 10.3934/dcds.1997.3.419

[2]

Y. A. Li, P. J. Olver. Convergence of solitary-wave solutions in a perturbed bi-hamiltonian dynamical system ii. complex analytic behavior and convergence to non-analytic solutions. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 159-191. doi: 10.3934/dcds.1998.4.159

[3]

Ernest Fontich, Pau Martín. Arnold diffusion in perturbations of analytic integrable Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 61-84. doi: 10.3934/dcds.2001.7.61

[4]

Alicia Cordero, José Martínez Alfaro, Pura Vindel. Bott integrable Hamiltonian systems on $S^{2}\times S^{1}$. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 587-604. doi: 10.3934/dcds.2008.22.587

[5]

Fuzhong Cong, Jialin Hong, Hongtian Li. Quasi-effective stability for nearly integrable Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 67-80. doi: 10.3934/dcdsb.2016.21.67

[6]

Sebastián Ferrer, Francisco Crespo. Parametric quartic Hamiltonian model. A unified treatment of classic integrable systems. Journal of Geometric Mechanics, 2014, 6 (4) : 479-502. doi: 10.3934/jgm.2014.6.479

[7]

Marcel Guardia. Splitting of separatrices in the resonances of nearly integrable Hamiltonian systems of one and a half degrees of freedom. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2829-2859. doi: 10.3934/dcds.2013.33.2829

[8]

Răzvan M. Tudoran, Anania Gîrban. On the Hamiltonian dynamics and geometry of the Rabinovich system. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 789-823. doi: 10.3934/dcdsb.2011.15.789

[9]

Marin Kobilarov, Jerrold E. Marsden, Gaurav S. Sukhatme. Geometric discretization of nonholonomic systems with symmetries. Discrete & Continuous Dynamical Systems - S, 2010, 3 (1) : 61-84. doi: 10.3934/dcdss.2010.3.61

[10]

Oscar E. Fernandez, Anthony M. Bloch, P. J. Olver. Variational Integrators for Hamiltonizable Nonholonomic Systems. Journal of Geometric Mechanics, 2012, 4 (2) : 137-163. doi: 10.3934/jgm.2012.4.137

[11]

Jorge Cortés, Manuel de León, Juan Carlos Marrero, Eduardo Martínez. Nonholonomic Lagrangian systems on Lie algebroids. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 213-271. doi: 10.3934/dcds.2009.24.213

[12]

José F. Cariñena, Irina Gheorghiu, Eduardo Martínez, Patrícia Santos. On the virial theorem for nonholonomic Lagrangian systems. Conference Publications, 2015, 2015 (special) : 204-212. doi: 10.3934/proc.2015.0204

[13]

Janusz Grabowski, Katarzyna Grabowska, Paweł Urbański. Geometry of Lagrangian and Hamiltonian formalisms in the dynamics of strings. Journal of Geometric Mechanics, 2014, 6 (4) : 503-526. doi: 10.3934/jgm.2014.6.503

[14]

Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of nonholonomic discrete mechanical systems. Journal of Geometric Mechanics, 2010, 2 (1) : 69-111. doi: 10.3934/jgm.2010.2.69

[15]

Francesco Fassò, Andrea Giacobbe, Nicola Sansonetto. On the number of weakly Noetherian constants of motion of nonholonomic systems. Journal of Geometric Mechanics, 2009, 1 (4) : 389-416. doi: 10.3934/jgm.2009.1.389

[16]

María Barbero-Liñán, Miguel C. Muñoz-Lecanda. Strict abnormal extremals in nonholonomic and kinematic control systems. Discrete & Continuous Dynamical Systems - S, 2010, 3 (1) : 1-17. doi: 10.3934/dcdss.2010.3.1

[17]

Dmitry V. Zenkov. Linear conservation laws of nonholonomic systems with symmetry. Conference Publications, 2003, 2003 (Special) : 967-976. doi: 10.3934/proc.2003.2003.967

[18]

Larry M. Bates, Francesco Fassò, Nicola Sansonetto. The Hamilton-Jacobi equation, integrability, and nonholonomic systems. Journal of Geometric Mechanics, 2014, 6 (4) : 441-449. doi: 10.3934/jgm.2014.6.441

[19]

Aristophanes Dimakis, Folkert Müller-Hoissen. Bidifferential graded algebras and integrable systems. Conference Publications, 2009, 2009 (Special) : 208-219. doi: 10.3934/proc.2009.2009.208

[20]

Leo T. Butler. A note on integrable mechanical systems on surfaces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1873-1878. doi: 10.3934/dcds.2014.34.1873

2017 Impact Factor: 0.561

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (15)

Other articles
by authors

[Back to Top]