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Lagrange-d'alembert-poincaré equations by several stages
On some aspects of the discretization of the suslov problem
Zentrum Mathematik der Technische Universität München, D-85747 Garching bei München, Germany |
In this paper we explore the discretization of Euler-Poincaré-Suslov equations on SO(3), i.e. of the Suslov problem. We show that the consistency order corresponding to the unreduced and reduced setups, when the discrete reconstruction equation is given by a Cayley retraction map, are related to each other in a nontrivial way. We give precise conditions under which general and variational integrators generate a discrete flow preserving the constraint distribution. We establish general consistency bounds and illustrate the performance of several discretizations by some plots. Moreover, along the lines of [
References:
[1] |
V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics; Dynamical Systems Ⅲ, Springer-Verlag, New York, 1989. Google Scholar |
[2] |
A. M. Bloch,
Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics Series 24, Springer-Verlag New-York, 2003. |
[3] |
A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden and R. Murray,
Nonholonomic mechanical systems with symmetry, Arch. Rational Mech. Anal., 136 (1996), 21-99.
doi: 10.1007/BF02199365. |
[4] |
A. I. Bobenko and Y. B. Suris,
Discrete lagrangian reduction, discrete Euler-Poincaré equations and semidirect products, Lett. Math. Phys., 49 (1999), 79-93.
doi: 10.1023/A:1007654605901. |
[5] |
N. Bou-Rabee and J. E. Marsden,
Hamilton-Pontryagin integrators on Lie groups: Introduction and structure-preserving properties, Foundations of Computational Mathematics, 9 (2009), 197-219.
doi: 10.1007/s10208-008-9030-4. |
[6] |
F. Cantrijn, M. de León, J. C. Marrero and D. Martín de Diego,
Reduction of nonholonomic mechanical systems with symmetry, Reports on Mathematical Physics, 42 (1998), 25-45.
doi: 10.1016/S0034-4877(98)80003-7. |
[7] |
J. Cortés and E. Martínez,
Nonholonomic integrators, Nonlinearity, 14 (2001), 1365-1392.
doi: 10.1088/0951-7715/14/5/322. |
[8] |
Y. N. Fedorov, A discretization of the nonholonomic Chaplygin sphere problem,
SIGMA: Symmetry Integrability Geom. Methods Appl. , 3 (2007), Paper 044, 15 pp. |
[9] |
Y. N. Fedorov and D. V. Zenkov,
Discrete nonholonomic LL systems on Lie groups, Nonlinearity, 18 (2005), 2211-2241.
doi: 10.1088/0951-7715/18/5/017. |
[10] |
S. Ferraro, D. Iglesias and D. Martín de Diego,
Momentum and energy preserving integrators for nonholonomic dynamics, Nonlinearity, 21 (2008), 1911-1928.
doi: 10.1088/0951-7715/21/8/009. |
[11] |
S. Ferraro, F. Jiménez and D. Martín de Diego,
New developments on the geometric nonholonomic integrator, Nonlinearity, 28 (2015), 871-900.
doi: 10.1088/0951-7715/28/4/871. |
[12] |
B. Fielder and J. Scheurle, Discretization of homoclinic orbits, rapid forcing and invisible chaos,
Memoirs of the American Mathematical Society, 119 (1996), viii+79 pp. |
[13] |
E. Hairer, C. Lubich and G. Wanner,
Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, 31, Springer-Verlag Berlin, 2002. |
[14] |
D. Iglesias, J. C. Marrero, D. Martín de Diego and E. Martínez,
Discrete nonholonomic Lagrangian systems on Lie groupoids, Journal of Nonlinear Sciences, 18 (2008), 351-397.
doi: 10.1007/s00332-007-9012-8. |
[15] |
F. Jiménez and J. Scheurle,
On the discretization of nonholonomic mechanics in ${{\mathbb{R}}^{n}}$, Journal of Geometric Mechanics, 7 (2015), 43-80.
doi: 10.3934/jgm.2015.7.43. |
[16] |
M. Kobilarov, D. Martín de Diego and S. Ferraro,
Ferraro, Simulating nonholonomic dynamics, Boletín de la Sociedad de Matemática Aplicada SeMA, 50 (2010), 61-81.
|
[17] |
V. V. Kozlov,
Invariant measures of the Euler-Poincaré equations on Lie algebras, Funct. Anal. Appl., 22 (1988), 58-59.
|
[18] |
M. de León,
A historical review on nonholonomic mechanics, Rev. R. Acad. Ciencias Exactas Fís. Nat. Serie A, 106 (2012), 191-224.
doi: 10.1007/s13398-011-0046-2. |
[19] |
J. E. Marsden, S. Pekarsky and S. Shkoller,
Discrete Euler-Poincaré and Lie-Poisson equations, Nonlinearity, 12 (1999), 1647-1662.
doi: 10.1088/0951-7715/12/6/314. |
[20] |
J. E. Marsden, S. Pekarsky and S. Shkoller,
Symmetry reduction of discrete Lagrangian mechanics on Lie groups, Journal of Geometry and Physics, 36 (2000), 140-151.
doi: 10.1016/S0393-0440(00)00018-8. |
[21] |
J. E. Marsden and M. West,
Discrete Mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.
doi: 10.1017/S096249290100006X. |
[22] |
R. McLachlan and M. Perlmutter,
Integrators for nonholonomic mechanical systems, J. Nonlinear Science, 16 (2006), 283-328.
doi: 10.1007/s00332-005-0698-1. |
[23] |
J. Moser and A. P. Veselov,
Discrete versions of some classical integrable systems and factorization of matrix polynomials, Comm. Math. Phys, 139 (1991), 217-243.
doi: 10.1007/BF02352494. |
[24] |
G. Suslov, Theoretical Mechanics, 2, Kiev (in Russian), 1902. Google Scholar |
[25] |
A. Weinstein,
Lagrangian mechanics and groupoids, Fields Inst. Comm., 7 (1996), 207-231.
|
show all references
References:
[1] |
V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics; Dynamical Systems Ⅲ, Springer-Verlag, New York, 1989. Google Scholar |
[2] |
A. M. Bloch,
Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics Series 24, Springer-Verlag New-York, 2003. |
[3] |
A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden and R. Murray,
Nonholonomic mechanical systems with symmetry, Arch. Rational Mech. Anal., 136 (1996), 21-99.
doi: 10.1007/BF02199365. |
[4] |
A. I. Bobenko and Y. B. Suris,
Discrete lagrangian reduction, discrete Euler-Poincaré equations and semidirect products, Lett. Math. Phys., 49 (1999), 79-93.
doi: 10.1023/A:1007654605901. |
[5] |
N. Bou-Rabee and J. E. Marsden,
Hamilton-Pontryagin integrators on Lie groups: Introduction and structure-preserving properties, Foundations of Computational Mathematics, 9 (2009), 197-219.
doi: 10.1007/s10208-008-9030-4. |
[6] |
F. Cantrijn, M. de León, J. C. Marrero and D. Martín de Diego,
Reduction of nonholonomic mechanical systems with symmetry, Reports on Mathematical Physics, 42 (1998), 25-45.
doi: 10.1016/S0034-4877(98)80003-7. |
[7] |
J. Cortés and E. Martínez,
Nonholonomic integrators, Nonlinearity, 14 (2001), 1365-1392.
doi: 10.1088/0951-7715/14/5/322. |
[8] |
Y. N. Fedorov, A discretization of the nonholonomic Chaplygin sphere problem,
SIGMA: Symmetry Integrability Geom. Methods Appl. , 3 (2007), Paper 044, 15 pp. |
[9] |
Y. N. Fedorov and D. V. Zenkov,
Discrete nonholonomic LL systems on Lie groups, Nonlinearity, 18 (2005), 2211-2241.
doi: 10.1088/0951-7715/18/5/017. |
[10] |
S. Ferraro, D. Iglesias and D. Martín de Diego,
Momentum and energy preserving integrators for nonholonomic dynamics, Nonlinearity, 21 (2008), 1911-1928.
doi: 10.1088/0951-7715/21/8/009. |
[11] |
S. Ferraro, F. Jiménez and D. Martín de Diego,
New developments on the geometric nonholonomic integrator, Nonlinearity, 28 (2015), 871-900.
doi: 10.1088/0951-7715/28/4/871. |
[12] |
B. Fielder and J. Scheurle, Discretization of homoclinic orbits, rapid forcing and invisible chaos,
Memoirs of the American Mathematical Society, 119 (1996), viii+79 pp. |
[13] |
E. Hairer, C. Lubich and G. Wanner,
Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, 31, Springer-Verlag Berlin, 2002. |
[14] |
D. Iglesias, J. C. Marrero, D. Martín de Diego and E. Martínez,
Discrete nonholonomic Lagrangian systems on Lie groupoids, Journal of Nonlinear Sciences, 18 (2008), 351-397.
doi: 10.1007/s00332-007-9012-8. |
[15] |
F. Jiménez and J. Scheurle,
On the discretization of nonholonomic mechanics in ${{\mathbb{R}}^{n}}$, Journal of Geometric Mechanics, 7 (2015), 43-80.
doi: 10.3934/jgm.2015.7.43. |
[16] |
M. Kobilarov, D. Martín de Diego and S. Ferraro,
Ferraro, Simulating nonholonomic dynamics, Boletín de la Sociedad de Matemática Aplicada SeMA, 50 (2010), 61-81.
|
[17] |
V. V. Kozlov,
Invariant measures of the Euler-Poincaré equations on Lie algebras, Funct. Anal. Appl., 22 (1988), 58-59.
|
[18] |
M. de León,
A historical review on nonholonomic mechanics, Rev. R. Acad. Ciencias Exactas Fís. Nat. Serie A, 106 (2012), 191-224.
doi: 10.1007/s13398-011-0046-2. |
[19] |
J. E. Marsden, S. Pekarsky and S. Shkoller,
Discrete Euler-Poincaré and Lie-Poisson equations, Nonlinearity, 12 (1999), 1647-1662.
doi: 10.1088/0951-7715/12/6/314. |
[20] |
J. E. Marsden, S. Pekarsky and S. Shkoller,
Symmetry reduction of discrete Lagrangian mechanics on Lie groups, Journal of Geometry and Physics, 36 (2000), 140-151.
doi: 10.1016/S0393-0440(00)00018-8. |
[21] |
J. E. Marsden and M. West,
Discrete Mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.
doi: 10.1017/S096249290100006X. |
[22] |
R. McLachlan and M. Perlmutter,
Integrators for nonholonomic mechanical systems, J. Nonlinear Science, 16 (2006), 283-328.
doi: 10.1007/s00332-005-0698-1. |
[23] |
J. Moser and A. P. Veselov,
Discrete versions of some classical integrable systems and factorization of matrix polynomials, Comm. Math. Phys, 139 (1991), 217-243.
doi: 10.1007/BF02352494. |
[24] |
G. Suslov, Theoretical Mechanics, 2, Kiev (in Russian), 1902. Google Scholar |
[25] |
A. Weinstein,
Lagrangian mechanics and groupoids, Fields Inst. Comm., 7 (1996), 207-231.
|


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