Geometric integrators for nonholonomic systems were introduced by Cortés and Martínez in [
Cortés and Martínez [
We evaluate the importance of the consistency condition by comparing the performance of two different geometric integrators for the nonholonomic Suslov problem, only one of which corresponds to a consistent choice of $L_d$ and $D_d$. We prove that both integrators produce approximations of the same order, and, moreover, that the non-consistent discretisation outperforms the other in numerical experiments and in terms of energy preservation. Our results indicate that the consistency of a discretisation might not be the most relevant feature to consider in the construction of nonholonomic geometric integrators.
Citation: |
Figure 2. The discrete momentum locus $\mathfrak{u}^{(\infty)}_{\varepsilon}$ defined by $\ell_d^{(\infty, \varepsilon)}$ and the plane $\mathfrak{d}^*$ immersed in $\mathfrak{so}(3)^*=\mathbb{R}^3$. They intersect along the red curve, which self-intersects at the origin where $\mathfrak{u}^{(\infty)}_{\varepsilon}$ is tangent to $\mathfrak{d}^*$.
Figure 3. Comparison between the two discretisations for a generic inertia tensor. The graph on the left shows the approximation of $M_1$ and the evolution of the energy $E_c$ (inset). The graph on the right shows the approximation of $M_2$ and the evolution of the signed distance $\rho$ to the constraint subspace $\mathfrak{d}^*$ (inset)
Figure 4. Comparison between of the two discretisations for a special inertia tensor having $I_{11}=I_{22}$. The graph on the left shows the approximation of $M_1$ and the evolution of the energy $E_c$ (inset). The graph on the right shows the approximation of $M_2$ and the evolution of the signed distance $\rho$ to the constraint subspace $\mathfrak{d}^*$ (inset)
[1] |
V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics; Dynamical Systems Ⅲ, 3rd edition, Encyclopaedia of Mathematical Sciences, 3. Springer-Verlag, Berlin, 2006.
doi: 10.1007/978-3-642-61551-1.![]() ![]() ![]() |
[2] |
A. M. Bloch, Nonholonomic Mechanics and Control, 2nd edition, Springer-Verlag, New York, 2015.
doi: 10.1007/b97376.![]() ![]() ![]() |
[3] |
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.![]() ![]() ![]() |
[4] |
J. Cortés and S. Martínez, Nonholonomic integrators, Nonlinearity, 14 (2001), 1365-1392.
doi: 10.1088/0951-7715/14/5/322.![]() ![]() ![]() |
[5] |
Y. N. Fedorov and V. V. Kozlov, Various aspects of n-dimensional rigid body dynamics, Amer. Math. Soc. Transl., 168 (1995), 141-171.
doi: 10.1090/trans2/168/06.![]() ![]() ![]() |
[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] |
Y. N. Fedorov, A discretization of the nonholonomic Chaplygin sphere problem, SIGMA, 3 (2007), Paper 044, 15pp.
doi: 10.3842/SIGMA.2007.044.![]() ![]() ![]() |
[8] |
Y. N. Fedorov, A. J. Maciejewski and M. Przybylska, The Poisson equations in the nonholonomic Suslov problem: Integrability, meromorphic and hypergeometric solutions, Nonlinearity, 22 (2009), 2231-2259.
doi: 10.1088/0951-7715/22/9/009.![]() ![]() ![]() |
[9] |
L. C. García-Naranjo, J. C. Marrero, A. J. Maciejewski and M. Przybylska, The inhomogeneous Suslov problem, Phys. Lett. A, 378 (2014), 2389-2394.
doi: 10.1016/j.physleta.2014.06.026.![]() ![]() ![]() |
[10] |
D. Iglesias, J. C. Marrero, D. Martín de Diego and E. Martínez, Discrete nonholonomic Lagrangian systems on Lie groupoids, J. Nonlinear Sci., 18 (2008), 351-397.
doi: 10.1007/s00332-007-9012-8.![]() ![]() ![]() |
[11] |
A. Iserles, H. Z. Munthe-Kaas, S. P. Norsett and A. Zanna, Lie-group methods, Acta Numerica., 9 (2000), 215-365.
doi: 10.1017/S0962492900002154.![]() ![]() ![]() |
[12] |
F. Jiménez and J. Scheurle, On the discretization of nonholonomic mechanics in $\mathbb{R}^N$, J. Geom. Mech., 7 (2015), 43-80.
doi: 10.3934/jgm.2015.7.43.![]() ![]() ![]() |
[13] |
F. Jiménez and J. Scheurle, On the discretization of the Euler-Poincaré-Suslov equations in SO(3), arXiv: 1506.01289. To appear in J. Geom. Mech.
![]() |
[14] |
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems, 2nd edition, Texts in Applied Mathematics, 17. SpringerVerlag, New York, 1999.
doi: 10.1007/978-0-387-21792-5.![]() ![]() ![]() |
[15] |
J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514.
doi: 10.1017/S096249290100006X.![]() ![]() ![]() |
[16] |
R. McLachlan and M. Perlmutter, Integrators for nonholonomic mechanical systems, J. Nonlinear Sci., 16 (2006), 283-328.
doi: 10.1007/s00332-005-0698-1.![]() ![]() ![]() |
[17] |
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.![]() ![]() ![]() |
[18] |
G. K. Suslov, Theoretical Mechanics, Gostekhizdat, Moscow, 1946 (in Russian).
![]() |
[19] |
A. P. Veselov, Integrable discrete-time systems and difference operators, Funct. Anal. Appl., 22 (1988), 1-13.
doi: 10.1007/bf01077598.![]() ![]() ![]() |