# American Institute of Mathematical Sciences

March  2018, 10(1): 43-68. doi: 10.3934/jgm.2018002

## On some aspects of the discretization of the suslov problem

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

Received  June 2015 Revised  October 2017 Published  December 2017

Fund Project: This research was supported by the DFG Collaborative Research Center TRR 109, "Discretization in Geometry and Dynamics".

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 [15] we show that any constraints-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.

Citation: Fernando Jiménez, Jürgen Scheurle. On some aspects of the discretization of the suslov problem. Journal of Geometric Mechanics, 2018, 10 (1) : 43-68. doi: 10.3934/jgm.2018002
##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] J. Cortés and E. Martínez, Nonholonomic integrators, Nonlinearity, 14 (2001), 1365-1392.  doi: 10.1088/0951-7715/14/5/322.  Google Scholar [8] Y. N. Fedorov, A discretization of the nonholonomic Chaplygin sphere problem, SIGMA: Symmetry Integrability Geom. Methods Appl. , 3 (2007), Paper 044, 15 pp.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  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, 31, Springer-Verlag Berlin, 2002.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [17] V. V. Kozlov, Invariant measures of the Euler-Poincaré equations on Lie algebras, Funct. Anal. Appl., 22 (1988), 58-59.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] J. E. Marsden and M. West, Discrete Mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.  doi: 10.1017/S096249290100006X.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] J. Cortés and E. Martínez, Nonholonomic integrators, Nonlinearity, 14 (2001), 1365-1392.  doi: 10.1088/0951-7715/14/5/322.  Google Scholar [8] Y. N. Fedorov, A discretization of the nonholonomic Chaplygin sphere problem, SIGMA: Symmetry Integrability Geom. Methods Appl. , 3 (2007), Paper 044, 15 pp.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  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, 31, Springer-Verlag Berlin, 2002.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [17] V. V. Kozlov, Invariant measures of the Euler-Poincaré equations on Lie algebras, Funct. Anal. Appl., 22 (1988), 58-59.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] J. E. Marsden and M. West, Discrete Mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.  doi: 10.1017/S096249290100006X.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar
In this figure we display the performance of the midpoint rule ($\overline{\mbox{DSP }}(\omega^k,\lambda_{k+1};\omega^{k+1}) = 0$, with inertia matrix $\mathbb{I}$ and initial values $\omega_1(0)$ and $\omega_2(0)$ introduced above) for the nonholonomic rigid body with a time step of size $\epsilon = 10^{-3}$. The solid red line is obtained through a RK4 integrator (which we consider an accurate approximation of the continuous nonlinear dynamics over a short time interval), while the blue dots represent the performance of the midpoint rule. The plots $(a)$ and $(b)$ correspond to the dynamical variables $\omega_1$, $\omega_2$, while $(c)$ displays the Lagrange multipliers $\lambda.$ On the other hand $(d)$ shows the inconsistent multipliers generated by the nonholonomic variational integrator. Finally, $(e)$ and $(f)$ show the preservation of the constraints and the energy $E_l(\hat\omega)$ up through round off errors, respectively.
This figure displays the comparison between the midpoint rule (the same as in Figure 1) and the variational integrator (37), (38), for a time step of size $\epsilon = 10^0 = 1$ (we recall that this integrator is also order 2 consistent in the dynamical variables). The former is represented by the green points and the latter by the blue ones, while the solid red line still represents the performance of a RK4 method. Variables $\omega_1$ $(a)$, $\omega_2$ $(b)$, $\lambda$ $(c)$ and $E_l$ $(d)$ are displayed, while $(e)$ shows the preservation of the constraints by the variational integrator up through round off errors. We observe a better performance of the variational integrator, mainly with respect to the preservation of energy, a fact which, considering bigger time steps, leads to the conclusion that its convergence to the actual solution is much faster and its long-term behavior is much more accurate.
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