On the discretization of nonholonomic dynamics in \mathbb{R}^n

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March 2015, 7(1): 43-80. doi: 10.3934/jgm.2015.7.43

On the discretization of nonholonomic dynamics in $\mathbb{R}^n$

Fernando Jiménez ^1, and Jürgen Scheurle ^1,

1.	Zentrum Mathematik der Technische Universität München, D-85747 Garching bei Munchen, Germany, Germany

Received July 2014 Revised February 2015 Published March 2015

Abstract
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In this paper we explore the nonholonomic Lagrangian setting of mechanical systems in coordinates on finite-dimensional configuration manifolds. We prove existence and uniqueness of solutions by reducing the basic equations of motion to a set of ordinary differential equations on the underlying distribution manifold $D$. Moreover, we show that any $D-$preserving discretization may be understood as being generated by the exact evolution map of a time-periodic non-autonomous perturbation of the original continuous-time nonholonomic system. By means of discretizing the corresponding Lagrange-d'Alembert principle, we construct geometric integrators for the original nonholonomic system. We give precise conditions under which these integrators generate a discrete flow preserving the distribution $D$. Also, we derive corresponding consistency estimates. Finally, we carefully treat the example of the nonholonomic particle, showing how to discretize the equations of motion in a reasonable way, particularly regarding the nonholonomic constraints.

Keywords: discrete variational calculus, Nonholonomic mechanics, geometric integration, differential algebraic equations., ordinary differential equations, discretization as perturbation.

Mathematics Subject Classification: 34C15, 37J15, 37N05, 65P10, 70F2.

Citation: Fernando Jiménez, Jürgen Scheurle. On the discretization of nonholonomic dynamics in $\mathbb{R}^n$. Journal of Geometric Mechanics, 2015, 7 (1) : 43-80. doi: 10.3934/jgm.2015.7.43

References:

[1]	A. M. Bloch, Nonholonomic Mechanics and Control,, Interdisciplinary Applied Mathematics Series, (2003). doi: 10.1007/b97376. Google Scholar
[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. doi: 10.1007/BF02199365. Google Scholar*
[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. doi: 10.1017/S0305004101005679. Google Scholar*
[4]	J. Cortés, Geometric, Control and Numerical Aspects of Nonholonomic Systems,, Lecture Notes in Mathematics, (1793). doi: 10.1007/b84020. Google Scholar
[5]	J. Cortés and E. Martínez, Nonholonomic integrators,, Nonlinearity, 14 (2001), 1365. Google Scholar
[6]	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. Google Scholar
[7]	S. Ferraro, D. Iglesias and D. Martín de Diego, Momentum and energy preserving integrators for nonholonomic dynamics,, Nonlinearity, 21 (2008), 1911. doi: 10.1088/0951-7715/21/8/009. Google Scholar
[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. Google Scholar
[9]	S. Ferraro, F. Jiménez and D. Martín de Diego, New developments on the geometric nonholonomic integrator,, accepted by Nonlinearity, (). Google Scholar
[10]	B. Fielder and J. Scheurle, Discretization of homoclinic orbits, rapid forcing and invisible chaos,, Memoirs of the American Mathematical Society, 119* (1996). doi: 10.1090/memo/0570. Google Scholar*
[11]	J. P. Fink and W. C. Rheinboldt, On the discretization error of parametrized nonlinear equations,, SIAM Journal on Numerical Analysis, 20 (1983), 732. doi: 10.1137/0720049. Google Scholar
[12]	Z. Ge and J. E. Marsden, Lie-Poisson integrators and Lie-Poisson Hamilton-Jacobi theory,, Phys. Lett. A., 133* (1988), 134. doi: 10.1016/0375-9601(88)90773-6. Google Scholar*
[13]	E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations,, Springer Series in Computational Mathematics, (2002). doi: 10.1007/978-3-662-05018-7. Google Scholar
[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. doi: 10.1007/s00332-007-9012-8. Google Scholar
[15]	Kobilarov M, D. Martín de Diego and S. Ferraro, Simulating nonholonomic dynamics,, Boletín de la Sociedad de Matemática Aplicada SeMA, 50 (2010), 61. Google Scholar
[16]	J. Koiller, Reduction of some classical nonholonomic systems with symmetry,, Arch. Rational Mech. Anal., 118* (1992), 113. doi: 10.1007/BF00375092. Google Scholar*
[17]	W. S. Koon and J. E. Marsden, Poisson reduction for nonholonomic mechanical systems with symmetry,, Rep. Math. Phys., 42 (1998), 101. doi: 10.1016/S0034-4877(98)80007-4. Google Scholar
[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. doi: 10.1007/s10883-007-9027-3. Google Scholar
[19]	M. de León and D. Martín de Diego, On the geometry of nonholonomic Lagrangian systems,, J. Math. Phys., 37 (1996), 3389. doi: 10.1063/1.531571. Google Scholar
[20]	R. McLachlan and M. Perlmutter, Integrators for nonholonomic mechanical systems,, J. Nonlinear Science, 16 (2006), 283. doi: 10.1007/s00332-005-0698-1. Google Scholar
[21]	J. E. Marsden and M. West, Discrete mechanics and variational integrators,, Acta Numerica, 10 (2001), 357. doi: 10.1017/S096249290100006X. Google Scholar
[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. doi: 10.1007/BF02352494. Google Scholar*
[23]	P. J. Rabier and W. C. Rheinboldt, A geometric treatment of implicit differential-algebraic equations,, Journal of Differential Equations, 109* (1994), 110. doi: 10.1006/jdeq.1994.1046. Google Scholar*
[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). Google Scholar
[25]	S. Reich, On an existence and uniqueness theory for nonlinear differential-algebraic equations,, Circuits, 11 (1992). doi: 10.1007/BF01189230. Google Scholar
[26]	S. Reich, On a geometrical interpretation of differential-algebraic equations,, Circuits, 10 (1991), 343. doi: 10.1007/BF01187550. Google Scholar
[27]	W. C. Rheinboldt, Differential-algebraic systems as differential equations on manifolds,, Mathematics of Computation, 43 (1984), 473. doi: 10.1090/S0025-5718-1984-0758195-5. Google Scholar

show all references

References:

[1]	A. M. Bloch, Nonholonomic Mechanics and Control,, Interdisciplinary Applied Mathematics Series, (2003). doi: 10.1007/b97376. Google Scholar
[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. doi: 10.1007/BF02199365. Google Scholar*
[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. doi: 10.1017/S0305004101005679. Google Scholar*
[4]	J. Cortés, Geometric, Control and Numerical Aspects of Nonholonomic Systems,, Lecture Notes in Mathematics, (1793). doi: 10.1007/b84020. Google Scholar
[5]	J. Cortés and E. Martínez, Nonholonomic integrators,, Nonlinearity, 14 (2001), 1365. Google Scholar
[6]	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. Google Scholar
[7]	S. Ferraro, D. Iglesias and D. Martín de Diego, Momentum and energy preserving integrators for nonholonomic dynamics,, Nonlinearity, 21 (2008), 1911. doi: 10.1088/0951-7715/21/8/009. Google Scholar
[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. Google Scholar
[9]	S. Ferraro, F. Jiménez and D. Martín de Diego, New developments on the geometric nonholonomic integrator,, accepted by Nonlinearity, (). Google Scholar
[10]	B. Fielder and J. Scheurle, Discretization of homoclinic orbits, rapid forcing and invisible chaos,, Memoirs of the American Mathematical Society, 119* (1996). doi: 10.1090/memo/0570. Google Scholar*
[11]	J. P. Fink and W. C. Rheinboldt, On the discretization error of parametrized nonlinear equations,, SIAM Journal on Numerical Analysis, 20 (1983), 732. doi: 10.1137/0720049. Google Scholar
[12]	Z. Ge and J. E. Marsden, Lie-Poisson integrators and Lie-Poisson Hamilton-Jacobi theory,, Phys. Lett. A., 133* (1988), 134. doi: 10.1016/0375-9601(88)90773-6. Google Scholar*
[13]	E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations,, Springer Series in Computational Mathematics, (2002). doi: 10.1007/978-3-662-05018-7. Google Scholar
[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. doi: 10.1007/s00332-007-9012-8. Google Scholar
[15]	Kobilarov M, D. Martín de Diego and S. Ferraro, Simulating nonholonomic dynamics,, Boletín de la Sociedad de Matemática Aplicada SeMA, 50 (2010), 61. Google Scholar
[16]	J. Koiller, Reduction of some classical nonholonomic systems with symmetry,, Arch. Rational Mech. Anal., 118* (1992), 113. doi: 10.1007/BF00375092. Google Scholar*
[17]	W. S. Koon and J. E. Marsden, Poisson reduction for nonholonomic mechanical systems with symmetry,, Rep. Math. Phys., 42 (1998), 101. doi: 10.1016/S0034-4877(98)80007-4. Google Scholar
[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. doi: 10.1007/s10883-007-9027-3. Google Scholar
[19]	M. de León and D. Martín de Diego, On the geometry of nonholonomic Lagrangian systems,, J. Math. Phys., 37 (1996), 3389. doi: 10.1063/1.531571. Google Scholar
[20]	R. McLachlan and M. Perlmutter, Integrators for nonholonomic mechanical systems,, J. Nonlinear Science, 16 (2006), 283. doi: 10.1007/s00332-005-0698-1. Google Scholar
[21]	J. E. Marsden and M. West, Discrete mechanics and variational integrators,, Acta Numerica, 10 (2001), 357. doi: 10.1017/S096249290100006X. Google Scholar
[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. doi: 10.1007/BF02352494. Google Scholar*
[23]	P. J. Rabier and W. C. Rheinboldt, A geometric treatment of implicit differential-algebraic equations,, Journal of Differential Equations, 109* (1994), 110. doi: 10.1006/jdeq.1994.1046. Google Scholar*
[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). Google Scholar
[25]	S. Reich, On an existence and uniqueness theory for nonlinear differential-algebraic equations,, Circuits, 11 (1992). doi: 10.1007/BF01189230. Google Scholar
[26]	S. Reich, On a geometrical interpretation of differential-algebraic equations,, Circuits, 10 (1991), 343. doi: 10.1007/BF01187550. Google Scholar
[27]	W. C. Rheinboldt, Differential-algebraic systems as differential equations on manifolds,, Mathematics of Computation, 43 (1984), 473. doi: 10.1090/S0025-5718-1984-0758195-5. Google Scholar

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