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Linear weakly Noetherian constants of motion are horizontal gauge momenta
Variational Integrators for Hamiltonizable Nonholonomic Systems
1. | Department of Mathematics, Wellesley College, Wellesley, MA 02482, USA Government |
2. | Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, United States |
3. | School of Mathematics, University of Minnesota, Minneapolis, MN 55455, United States |
References:
[1] |
A. M. Bloch, "Nonholonomic Mechanics and Control,'', Interdisciplinary Applied Mathematics, 24 (2003).
|
[2] |
A. M. Bloch, O. E. Fernandez and T. Mestdag, Hamiltonization of nonholonomic systems and the inverse problem of the calculus of variations,, Rep. Math. Phys., 63 (2009), 225.
doi: 10.1016/S0034-4877(09)90001-5. |
[3] |
A. V. Borisov and I. S. Mamaev, Rolling of a rigid body on plane and sphere. Hierarchy of dynamics,, Reg. Chaotic Dyn., 7 (2002), 177.
|
[4] |
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. |
[5] |
R. L. Burden and J. D. Faires, "Numerical Analysis,'', 8th edition, (2005). Google Scholar |
[6] |
S. A. Chaplygin, On a ball's rolling on a horizontal plane,, (in Russian), 24 (1903), 139.
|
[7] |
S. A. Chaplygin, On the theory of motion of nonholonomic systems. The reducing-multiplier theorem,, (in Russian), 28 (1911), 303.
|
[8] |
J. Cortés Monforte, "Geometric, Control and Numerical Aspects of Nonholonomic Systems,'', Lecture Notes in Mathematics, 1793 (2002).
|
[9] |
J. Cortés Monforte and S. Martĺnez, Nonholonomic integrators,, Nonlinearity, 14 (2001), 1365.
doi: 10.1088/0951-7715/14/5/322. |
[10] |
Y. N. Fedorov and B. Jovanović, Quasi-Chaplygin systems and nonholonomic rigid body dynamics,, Lett. Math. Phys., 76 (2006), 215.
doi: 10.1007/s11005-006-0069-3. |
[11] |
Y. N. Fedorov and D. V. Zenkov, Discrete nonholonomic LL systems on Lie Groups,, Nonlinearity, 18 (2005), 2211.
doi: 10.1088/0951-7715/18/5/017. |
[12] |
O. E. Fernandez, "The Hamiltonization of Nonholonomic Systems and its Applications,'', Ph.D. Thesis, (2009).
|
[13] |
O. E. Fernandez and A. M. Bloch, The Weitzenböck connection and time reparameterization in nonholonomic mechanics,, J. Math. Phys., 52 (2011).
doi: 10.1063/1.3525798. |
[14] |
O. E. Fernandez and A. M. Bloch, Equivalence of the dynamics of nonholonomic and variational nonholonomic systems for certain initial data,, J. Phys. A, 41 (2008).
doi: 10.1088/1751-8113/41/34/344005. |
[15] |
O. E. Fernandez, T. Mestdag and A. M. Bloch, A generalization of Chaplygin's reducibility theorem,, Reg. Chaotic Dyn., 14 (2009), 635.
doi: 10.1134/S1560354709060033. |
[16] |
P. Fitzpatrick, "Advanced Calculus,'', 2nd edition, (2006). Google Scholar |
[17] |
M. R. Flannery, The enigma of nonholonomic constraints,, Am. J. of Phys., 73 (2005), 265.
doi: 10.1119/1.1830501. |
[18] |
Z. Ge and J. E. Marsden, Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators,, Phys. Lett. A, 133 (1988), 134.
doi: 10.1016/0375-9601(88)90773-6. |
[19] |
E. Hairer, Variable time step integration with symplectic methods,, Appl. Numer. Math., 25 (1997), 219.
doi: 10.1016/S0168-9274(97)00061-5. |
[20] |
D. Iglesias, J. C. Marrero, D. M. de Diego and E. Martínez, Discrete nonholonomic Lagrangian systems on Lie groupoids,, J. Nonlinear Sci., 18 (2008), 221.
doi: 10.1007/s00332-007-9012-8. |
[21] |
M. Kobilarov, J. E. Marsden and G. S. Sukhatme, Geometric discretization of nonholonomic systems with symmetries,, Discrete and Continuous Dynamical Systems, 3 (2010), 61.
|
[22] |
D. Korteweg, Über eine ziemlich verbreitete unrichtige Behandlungsweise eines Problemes der rollenden Bewegung und insbesondere Über kleine rollende Schwingungen um eine Gleichgewichtslage,, Nieuw Archiefvoor Wiskunde, 4 (1899), 130. Google Scholar |
[23] |
B. Leimkuhler and S. Reich, "Simulating Hamiltonian Dynamics,'', Cambridge Monographs on Applied and Computational Mathematics, 14 (2004).
|
[24] |
B. Leimkuhler and R. Skeel, Symplectic numerical integrators in constrained Hamiltonian systems,, J. Computational Phys., 112 (1994), 117.
doi: 10.1006/jcph.1994.1085. |
[25] |
M. Leok and J. Zhang, Discrete Hamiltonian variational integrators,, IMA J. Numerical Analysis, 31 (2011), 1497.
|
[26] |
J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems,'', 2nd edition, 17 (1999).
|
[27] |
J. E. Marsden and M. West, Discrete mechanics and variational integrators,, Acta Numerica, 10 (2001), 357.
doi: 10.1017/S096249290100006X. |
[28] |
R. McLachlan and M. Perlmutter, Integrators for nonholonomic mechanical systems,, J. Nonlinear Sci., 16 (2006), 283.
doi: 10.1007/s00332-005-0698-1. |
[29] |
T. Mestdag, A. M. Bloch and O. E. Fernandez, Hamiltonization and geometric integration of nonholonomic mechanical systems,, in, (2009), 230. Google Scholar |
[30] |
F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, eds., "NIST Handbook of Mathematical Functions,'', U.S. Department of Commerce, (2010).
|
[31] |
T. Ohsawa, O. E. Fernandez, A. M. Bloch and D. V. Zenkov, Nonholonomic Hamilton-Jacobi theory via Chaplygin Hamiltonization,, J. Geometry and Phys., 61 (2011), 1263.
doi: 10.1016/j.geomphys.2011.02.015. |
[32] |
J. Ryckaert, G. Ciccotti and H. Berendsen, Numerical integration of the cartesian equations of motion of a system with constraints: Molecular dynamics of n-alkanes,, J. Computational Phs., 23 (1977), 327.
doi: 10.1016/0021-9991(77)90098-5. |
[33] |
B. van Brunt, "The Calculus of Variations,'', Universitext, (2004).
|
[34] |
L. Verlet, Computer experiments on classical fluids,, Phys. Rev., 159 (1967), 98.
doi: 10.1103/PhysRev.159.98. |
show all references
References:
[1] |
A. M. Bloch, "Nonholonomic Mechanics and Control,'', Interdisciplinary Applied Mathematics, 24 (2003).
|
[2] |
A. M. Bloch, O. E. Fernandez and T. Mestdag, Hamiltonization of nonholonomic systems and the inverse problem of the calculus of variations,, Rep. Math. Phys., 63 (2009), 225.
doi: 10.1016/S0034-4877(09)90001-5. |
[3] |
A. V. Borisov and I. S. Mamaev, Rolling of a rigid body on plane and sphere. Hierarchy of dynamics,, Reg. Chaotic Dyn., 7 (2002), 177.
|
[4] |
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. |
[5] |
R. L. Burden and J. D. Faires, "Numerical Analysis,'', 8th edition, (2005). Google Scholar |
[6] |
S. A. Chaplygin, On a ball's rolling on a horizontal plane,, (in Russian), 24 (1903), 139.
|
[7] |
S. A. Chaplygin, On the theory of motion of nonholonomic systems. The reducing-multiplier theorem,, (in Russian), 28 (1911), 303.
|
[8] |
J. Cortés Monforte, "Geometric, Control and Numerical Aspects of Nonholonomic Systems,'', Lecture Notes in Mathematics, 1793 (2002).
|
[9] |
J. Cortés Monforte and S. Martĺnez, Nonholonomic integrators,, Nonlinearity, 14 (2001), 1365.
doi: 10.1088/0951-7715/14/5/322. |
[10] |
Y. N. Fedorov and B. Jovanović, Quasi-Chaplygin systems and nonholonomic rigid body dynamics,, Lett. Math. Phys., 76 (2006), 215.
doi: 10.1007/s11005-006-0069-3. |
[11] |
Y. N. Fedorov and D. V. Zenkov, Discrete nonholonomic LL systems on Lie Groups,, Nonlinearity, 18 (2005), 2211.
doi: 10.1088/0951-7715/18/5/017. |
[12] |
O. E. Fernandez, "The Hamiltonization of Nonholonomic Systems and its Applications,'', Ph.D. Thesis, (2009).
|
[13] |
O. E. Fernandez and A. M. Bloch, The Weitzenböck connection and time reparameterization in nonholonomic mechanics,, J. Math. Phys., 52 (2011).
doi: 10.1063/1.3525798. |
[14] |
O. E. Fernandez and A. M. Bloch, Equivalence of the dynamics of nonholonomic and variational nonholonomic systems for certain initial data,, J. Phys. A, 41 (2008).
doi: 10.1088/1751-8113/41/34/344005. |
[15] |
O. E. Fernandez, T. Mestdag and A. M. Bloch, A generalization of Chaplygin's reducibility theorem,, Reg. Chaotic Dyn., 14 (2009), 635.
doi: 10.1134/S1560354709060033. |
[16] |
P. Fitzpatrick, "Advanced Calculus,'', 2nd edition, (2006). Google Scholar |
[17] |
M. R. Flannery, The enigma of nonholonomic constraints,, Am. J. of Phys., 73 (2005), 265.
doi: 10.1119/1.1830501. |
[18] |
Z. Ge and J. E. Marsden, Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators,, Phys. Lett. A, 133 (1988), 134.
doi: 10.1016/0375-9601(88)90773-6. |
[19] |
E. Hairer, Variable time step integration with symplectic methods,, Appl. Numer. Math., 25 (1997), 219.
doi: 10.1016/S0168-9274(97)00061-5. |
[20] |
D. Iglesias, J. C. Marrero, D. M. de Diego and E. Martínez, Discrete nonholonomic Lagrangian systems on Lie groupoids,, J. Nonlinear Sci., 18 (2008), 221.
doi: 10.1007/s00332-007-9012-8. |
[21] |
M. Kobilarov, J. E. Marsden and G. S. Sukhatme, Geometric discretization of nonholonomic systems with symmetries,, Discrete and Continuous Dynamical Systems, 3 (2010), 61.
|
[22] |
D. Korteweg, Über eine ziemlich verbreitete unrichtige Behandlungsweise eines Problemes der rollenden Bewegung und insbesondere Über kleine rollende Schwingungen um eine Gleichgewichtslage,, Nieuw Archiefvoor Wiskunde, 4 (1899), 130. Google Scholar |
[23] |
B. Leimkuhler and S. Reich, "Simulating Hamiltonian Dynamics,'', Cambridge Monographs on Applied and Computational Mathematics, 14 (2004).
|
[24] |
B. Leimkuhler and R. Skeel, Symplectic numerical integrators in constrained Hamiltonian systems,, J. Computational Phys., 112 (1994), 117.
doi: 10.1006/jcph.1994.1085. |
[25] |
M. Leok and J. Zhang, Discrete Hamiltonian variational integrators,, IMA J. Numerical Analysis, 31 (2011), 1497.
|
[26] |
J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems,'', 2nd edition, 17 (1999).
|
[27] |
J. E. Marsden and M. West, Discrete mechanics and variational integrators,, Acta Numerica, 10 (2001), 357.
doi: 10.1017/S096249290100006X. |
[28] |
R. McLachlan and M. Perlmutter, Integrators for nonholonomic mechanical systems,, J. Nonlinear Sci., 16 (2006), 283.
doi: 10.1007/s00332-005-0698-1. |
[29] |
T. Mestdag, A. M. Bloch and O. E. Fernandez, Hamiltonization and geometric integration of nonholonomic mechanical systems,, in, (2009), 230. Google Scholar |
[30] |
F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, eds., "NIST Handbook of Mathematical Functions,'', U.S. Department of Commerce, (2010).
|
[31] |
T. Ohsawa, O. E. Fernandez, A. M. Bloch and D. V. Zenkov, Nonholonomic Hamilton-Jacobi theory via Chaplygin Hamiltonization,, J. Geometry and Phys., 61 (2011), 1263.
doi: 10.1016/j.geomphys.2011.02.015. |
[32] |
J. Ryckaert, G. Ciccotti and H. Berendsen, Numerical integration of the cartesian equations of motion of a system with constraints: Molecular dynamics of n-alkanes,, J. Computational Phs., 23 (1977), 327.
doi: 10.1016/0021-9991(77)90098-5. |
[33] |
B. van Brunt, "The Calculus of Variations,'', Universitext, (2004).
|
[34] |
L. Verlet, Computer experiments on classical fluids,, Phys. Rev., 159 (1967), 98.
doi: 10.1103/PhysRev.159.98. |
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