“A symmetric splitting method for rigid body dynamics”
Authors: E. Celledoni and N. Säfström,Affiliation: NTNU
Reference: 2006, Vol 27, No 2, pp. 95-108.
Keywords: Rigid bodies, symmetric methods, symplectic methods, Jacobi elliptic functions
Abstract: It has been known since the time of Jacobi that the solution to the free rigid body (FRB) equations of motion is given in terms of a certain type of elliptic functions. Using the Arithmetic-Geometric mean algorithm, (1), these functions can be calculated efficiently and accurately. The overall approach yields a faster and more accurate numerical solution to the FRB equations compared to standard numerical ODE and symplectic solvers. In this paper we investigate the possibility of extending this approach to the case of rigid bodies subject to external forces. By using a splitting strategy similar to the one proposed in (14), we decompose the vector field of our problem in a FRB problem and another completely integrable vector field. We apply the method to the simulation of the heavy top.
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DOI: 10.4173/mic.2006.2.2DOI forward links to this article:
| [1] Mechthild Thalhammer (2008), doi:10.1137/060674636 |
| [2] Alexey V. Akimov and Anatoly B. Kolomeisky (2011), doi:10.1021/ct200334e |
[1] Dover, Publications. (1992). Handbook of mathematical functions with formulas, graphs, and mathematical tables, Volume 55. Reprint of the 1972 edition. Dover Publications, Inc., New York.
[2] McLachlan, R. A. Dullweber, B. Leimkuhler. (1997). Symplectic splitting methods for rigid body molecular dynamics, J. Chem. Phys., 107:5840-5851 doi:10.1063/1.474310
[3] Butcher, J., (2003). Numerical Methods for Ordinary Differential Equations, Wiley, second edition edition.
[4] Celledoni, E. B. Owren. (2003). Lie group methods for rigid body dynamics and time integration on manifolds, Comput. Methods Appl. Mech. Engrg., 192:421-438 doi:10.1016/S0045-7825(02)00520-0
[5] Ge, Z. J.E. Marsden. (1988). Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators, Physics Letters A, 133(3):134-139 doi:10.1016/0375-9601(88)90773-6
[6] Geradin, M., A. Cardona. (2001). Flexible Multibody Dynamics, Wiley and Sons Ltd.
[7] Hairer, E., C. Lubich, G. Wanner. (2002). Geometric numerical integration, Volume 31 of Springer series in compuitational mathematics. Springer.
[8] Leimkuhler, B., S. Reich. (2004). Simulating Hamiltonian Dynamics, Volume 14 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, first edition edition.
[9] Lewis, D., J. C. Simo. (1994). Conserving algorithms for the dynamics of Hamiltonian systems of Lie groups, J. Nonlinear Sci., 4:253-299 doi:10.1007/BF02430634
[10] McLachlan, R. I., (1993). Explicit Lie-Poisson integration and the Euler equations, Physical Review Letters, 71:3043-3046 doi:10.1103/PhysRevLett.71.3043
[11] McLachlan, R. I., A. Zanna. (2005). The discrete Moser-Veselov algorithm for the free rigid body, revisited, Found. of Comp. Math., .1:87-123 doi:10.1007/s10208-004-0118-6
[12] Mitchell, J.Wm., (2000). A simplified variation of parameters solution for the motion of an arbirarily torqued mass asymmetric rigid body, PhD thesis, University of Cincinnati.
[13] Moser, J., A. Veselov. (1991). Discrete versions of some classical integrable systems and factorization of matrix polynomials, J. of Comm. Math. Phys., 13.2:217-243 doi:10.1007/BF02352494
[14] Reich, S., (1996). Symplectic integrators for systems of rigid bodies, Integration algorithms and classical mechanics, Fields Inst. Commun., Vol. 10, 181-191.
BibTeX:
@article{MIC-2006-2-2,
title={{A symmetric splitting method for rigid body dynamics}},
author={Celledoni, E. and Säfström, N.},
journal={Modeling, Identification and Control},
volume={27},
number={2},
pages={95--108},
year={2006},
doi={10.4173/mic.2006.2.2},
publisher={Norwegian Society of Automatic Control}
};