September  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).   Google Scholar

[2]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[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.  Google Scholar

[29]

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

[30]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[35]

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

[36]

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

[37]

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

[38]

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

[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.  Google Scholar

[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.  Google Scholar

[41]

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

[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.  Google Scholar

[43]

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

[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.  Google Scholar

[45]

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

[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.   Google Scholar

[47]

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

[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.  Google Scholar

[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.  Google Scholar

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).   Google Scholar

[2]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[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.  Google Scholar

[29]

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

[30]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[35]

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

[36]

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

[37]

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

[38]

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

[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.  Google Scholar

[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.  Google Scholar

[41]

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

[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.  Google Scholar

[43]

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

[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.  Google Scholar

[45]

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

[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.   Google Scholar

[47]

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

[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.  Google Scholar

[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.  Google Scholar

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