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On the discretization of nonholonomic dynamics in $\mathbb{R}^n$
1. | Zentrum Mathematik der Technische Universität München, D-85747 Garching bei Munchen, Germany, Germany |
References:
[1] |
A. M. Bloch, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics Series, 24, Springer-Verlag, New York, 2003.
doi: 10.1007/b97376. |
[2] |
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. |
[3] |
F. Cantrijn, J. Cortés, M. de León and D. Martín de Diego, On the geometry of generalized Chaplygin systems, Math. Proc. Cambridge Philos. Soc., 132 (2002), 323-351.
doi: 10.1017/S0305004101005679. |
[4] |
J. Cortés, Geometric, Control and Numerical Aspects of Nonholonomic Systems, Lecture Notes in Mathematics, 1793, Springer-Verlag, Berlin, 2002.
doi: 10.1007/b84020. |
[5] |
J. Cortés and E. Martínez, Nonholonomic integrators, Nonlinearity, 14 (2001), 1365-1392. |
[6] |
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. |
[7] |
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. |
[8] |
S. Ferraro, D. Iglesias and D. Martín de Diego, Numerical and geometric aspects of the nonholonomic SHAKE and RATTLE methods, Discrete and Continuous Dynamical Systems, (2009), 220-229. |
[9] |
S. Ferraro, F. Jiménez and D. Martín de Diego, New developments on the geometric nonholonomic integrator, accepted by Nonlinearity, arXiv:1312.1587. |
[10] |
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.
doi: 10.1090/memo/0570. |
[11] |
J. P. Fink and W. C. Rheinboldt, On the discretization error of parametrized nonlinear equations, SIAM Journal on Numerical Analysis, 20 (1983), 732-746.
doi: 10.1137/0720049. |
[12] |
Z. Ge and J. E. Marsden, Lie-Poisson integrators and Lie-Poisson Hamilton-Jacobi theory, Phys. Lett. A., 133 (1988), 134-139.
doi: 10.1016/0375-9601(88)90773-6. |
[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.
doi: 10.1007/978-3-662-05018-7. |
[14] |
D. Iglesias, J. C. Marrero, Martín de Diego and E. Martínez, Discrete nonholonomic Lagrangian systems on Lie groupoids, Journal of Nonlinear Sciences, 18 (2008), 221-276.
doi: 10.1007/s00332-007-9012-8. |
[15] |
Kobilarov M, D. Martín de Diego and S. Ferraro, Simulating nonholonomic dynamics, Boletín de la Sociedad de Matemática Aplicada SeMA, 50 (2010), 61-81. |
[16] |
J. Koiller, Reduction of some classical nonholonomic systems with symmetry, Arch. Rational Mech. Anal., 118 (1992), 113-148.
doi: 10.1007/BF00375092. |
[17] |
W. S. Koon and J. E. Marsden, Poisson reduction for nonholonomic mechanical systems with symmetry, Rep. Math. Phys., 42 (1998), 101-134.
doi: 10.1016/S0034-4877(98)80007-4. |
[18] |
K. Hüper and F. Silva Leite, On the geometry of rolling and interpolating curves on $S^n$, $SO_n$ and Grassman manifolds, Journal of Dynamical and Control Systems, 13 (2007), 467-502.
doi: 10.1007/s10883-007-9027-3. |
[19] |
M. de León and D. Martín de Diego, On the geometry of nonholonomic Lagrangian systems, J. Math. Phys., 37 (1996), 3389-3414.
doi: 10.1063/1.531571. |
[20] |
R. McLachlan and M. Perlmutter, Integrators for nonholonomic mechanical systems, J. Nonlinear Science, 16 (2006), 283-328.
doi: 10.1007/s00332-005-0698-1. |
[21] |
J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.
doi: 10.1017/S096249290100006X. |
[22] |
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. |
[23] |
P. J. Rabier and W. C. Rheinboldt, A geometric treatment of implicit differential-algebraic equations, Journal of Differential Equations, 109 (1994), 110-146.
doi: 10.1006/jdeq.1994.1046. |
[24] |
P. J. Rabier and W. C. Rheinboldt, Nonholonomic Motion of Rigid Mechanical Systems from a DAE Viewpoint, Society for Industrial and Applied Mathematics, 1987. |
[25] |
S. Reich, On an existence and uniqueness theory for nonlinear differential-algebraic equations, Circuits, Systems and Signal Processing, 11 (1992), p281.
doi: 10.1007/BF01189230. |
[26] |
S. Reich, On a geometrical interpretation of differential-algebraic equations, Circuits, Systems and Signal Processing, 10 (1991), 343-359.
doi: 10.1007/BF01187550. |
[27] |
W. C. Rheinboldt, Differential-algebraic systems as differential equations on manifolds, Mathematics of Computation, 43 (1984), 473-482.
doi: 10.1090/S0025-5718-1984-0758195-5. |
show all references
References:
[1] |
A. M. Bloch, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics Series, 24, Springer-Verlag, New York, 2003.
doi: 10.1007/b97376. |
[2] |
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. |
[3] |
F. Cantrijn, J. Cortés, M. de León and D. Martín de Diego, On the geometry of generalized Chaplygin systems, Math. Proc. Cambridge Philos. Soc., 132 (2002), 323-351.
doi: 10.1017/S0305004101005679. |
[4] |
J. Cortés, Geometric, Control and Numerical Aspects of Nonholonomic Systems, Lecture Notes in Mathematics, 1793, Springer-Verlag, Berlin, 2002.
doi: 10.1007/b84020. |
[5] |
J. Cortés and E. Martínez, Nonholonomic integrators, Nonlinearity, 14 (2001), 1365-1392. |
[6] |
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. |
[7] |
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. |
[8] |
S. Ferraro, D. Iglesias and D. Martín de Diego, Numerical and geometric aspects of the nonholonomic SHAKE and RATTLE methods, Discrete and Continuous Dynamical Systems, (2009), 220-229. |
[9] |
S. Ferraro, F. Jiménez and D. Martín de Diego, New developments on the geometric nonholonomic integrator, accepted by Nonlinearity, arXiv:1312.1587. |
[10] |
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.
doi: 10.1090/memo/0570. |
[11] |
J. P. Fink and W. C. Rheinboldt, On the discretization error of parametrized nonlinear equations, SIAM Journal on Numerical Analysis, 20 (1983), 732-746.
doi: 10.1137/0720049. |
[12] |
Z. Ge and J. E. Marsden, Lie-Poisson integrators and Lie-Poisson Hamilton-Jacobi theory, Phys. Lett. A., 133 (1988), 134-139.
doi: 10.1016/0375-9601(88)90773-6. |
[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.
doi: 10.1007/978-3-662-05018-7. |
[14] |
D. Iglesias, J. C. Marrero, Martín de Diego and E. Martínez, Discrete nonholonomic Lagrangian systems on Lie groupoids, Journal of Nonlinear Sciences, 18 (2008), 221-276.
doi: 10.1007/s00332-007-9012-8. |
[15] |
Kobilarov M, D. Martín de Diego and S. Ferraro, Simulating nonholonomic dynamics, Boletín de la Sociedad de Matemática Aplicada SeMA, 50 (2010), 61-81. |
[16] |
J. Koiller, Reduction of some classical nonholonomic systems with symmetry, Arch. Rational Mech. Anal., 118 (1992), 113-148.
doi: 10.1007/BF00375092. |
[17] |
W. S. Koon and J. E. Marsden, Poisson reduction for nonholonomic mechanical systems with symmetry, Rep. Math. Phys., 42 (1998), 101-134.
doi: 10.1016/S0034-4877(98)80007-4. |
[18] |
K. Hüper and F. Silva Leite, On the geometry of rolling and interpolating curves on $S^n$, $SO_n$ and Grassman manifolds, Journal of Dynamical and Control Systems, 13 (2007), 467-502.
doi: 10.1007/s10883-007-9027-3. |
[19] |
M. de León and D. Martín de Diego, On the geometry of nonholonomic Lagrangian systems, J. Math. Phys., 37 (1996), 3389-3414.
doi: 10.1063/1.531571. |
[20] |
R. McLachlan and M. Perlmutter, Integrators for nonholonomic mechanical systems, J. Nonlinear Science, 16 (2006), 283-328.
doi: 10.1007/s00332-005-0698-1. |
[21] |
J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.
doi: 10.1017/S096249290100006X. |
[22] |
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. |
[23] |
P. J. Rabier and W. C. Rheinboldt, A geometric treatment of implicit differential-algebraic equations, Journal of Differential Equations, 109 (1994), 110-146.
doi: 10.1006/jdeq.1994.1046. |
[24] |
P. J. Rabier and W. C. Rheinboldt, Nonholonomic Motion of Rigid Mechanical Systems from a DAE Viewpoint, Society for Industrial and Applied Mathematics, 1987. |
[25] |
S. Reich, On an existence and uniqueness theory for nonlinear differential-algebraic equations, Circuits, Systems and Signal Processing, 11 (1992), p281.
doi: 10.1007/BF01189230. |
[26] |
S. Reich, On a geometrical interpretation of differential-algebraic equations, Circuits, Systems and Signal Processing, 10 (1991), 343-359.
doi: 10.1007/BF01187550. |
[27] |
W. C. Rheinboldt, Differential-algebraic systems as differential equations on manifolds, Mathematics of Computation, 43 (1984), 473-482.
doi: 10.1090/S0025-5718-1984-0758195-5. |
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