LIE-POISSON INTEGRATORS FOR A RIGID SATELLITE ON A CIRCULAR ORBIT

Abstract

In the last two decades, many structure preserving numerical methods like Poisson integrators have been investigated in numerical studies. Since the structure matrices are different in many Poisson systems, no integrator is known yet to preserve the Poisson structure of any Poisson system. In the present paper, we propose Lie– Poisson integrators for Lie–Poisson systems whose structure matrix is different from the ones studied before. In particular, explicit Lie-Poisson integrators for the equations of rotational motion of a rigid body the satellite on a circular orbit around a fixed gravitational center have been constructed based on the splitting. The splitted parts have been composed by a first, a second and a third order compositions. It has been shown that the proposed schemes preserve the quadratic invariants of the equation. Numerical results reveal the preservation of the energy and agree with the theoretical treatment that the invariants lie on the sphere in long–term with different orders of accuracy

Keywords

• Lie-Poisson integrators,symplectic integrators,rigid body equations,splittings

References

1. Hairer, E., Lubich, C. and Wanner, G., (1987), Geometric Numerical Integration Structure–Preserving Algorithms for Ordinary Differential Equations, Springer (31), New York.
2. Antone, V. and Gladwell, I., (2004), Performance of Variable Step Size Methods for Solving Model Separable Hamiltonian Systems, Mathematical and Computer Modelling, 40, 1245–1262.
3. Tocino, A. and Vigo–Aguiar, J., (2005), Symplectic Conditions for Exponential Fitting Runge–Kutta– Nystr¨om Methods, Mathematical and Computer Modelling, 42, 873–876.
4. Reich, S., (1996), Symplectic Integrators for Systems of Rigid Bodies, In Integration algorithms and classical mechanics, Fields Inst. Commun. 10, Amer. Math. Soc., 181–191.
5. Channell, P.J. and Scovel, J.C., (1991), Integrators for Lie–Poisson dynamical systems, Physica D, 50, 80–88.
6. Ge, Z.G. and Marsden, J.E., (1998), Lie–Poisson–Hamiltonian theory and Lie–Poisson integrators. Physics Letter A. 133(3), 134–139.
7. McLachlan, R.I., (1993), Explicit Lie–Poisson Integration and the Euler Equations. Physics Review E. 71, 3043–3046.
8. Breiter, S., Nesvorn´y, D. and Vokrouhlick´y, D., (2005), Efficient Lie–Poisson Integrator for Secular Spin Dynamics of Rigid Bodies. The Astronomical Journal, 130, 1267–1277.

9. Austin, M.A., Krishnaprasad, P.S. and Wang, L.S., (1993), Almost Poisson Intagration of Rigid Body System. Journal of Computational Physics. 107, 105–117.
10. Ergen¸c, T. and Karas¨ozen, B., (2006), Poisson integrators for Volterra lattice equations. Applied Numerical Mathematics. 56, 879–887.
11. Jay, L.O., (2004), Preserving poisson structure and orthogonality in numerical integration of differen- tial equations. Computers and Mathematics with Applications. 48, 237–255.
12. Aydın, A., (1998), Poisson Integrators for Completely Integrable Hamiltonian Systems. M.Sc. thesis, Department of Mathematics, Middle East Technical University, Ankara.
13. Dullin, H.R., (2004), Poisson integrator for Symmetric Rigid Bodies. Regular and Chaotic Dynamics, 9(3), 255–264.
14. Karas¨ozen, B., (2004), Poisson integrators. Mathematical and Computer Modelling, 40, 1225–1244.
15. Aydın, A. and Karas¨ozen, B., (2007), Symplectic and multi-symplectic methods for coupled nonlinear Schrdinger equations with periodic solutions. Computer Physics Communications, 177, 566–583.
16. Aydın, A. and Karas¨ozen, B., (2008), Symplectic and multisymplectic Lobatto methods for the good Boussinesq equation. Journal of Mathematical Physics, 49, 083509(1)–18.
17. Aydın, A., (2009), Multisymplectic integration of N–coupled nonlinear Schr¨odinger equation with destabilized periodic wave solutions, Chaos, Solitons and Fractals, 41, 735751.
18. Leimkuhler, B. and Reich, S., (2004), Simulating Hamitonian Dynamics, Cambridge University Press:Cambridge.
19. Bhardwaj, R. and Kaur, P., (2006), Satellite’s Motion under the Effect of Magnetic Torque. American Journal of Applied Sciences, 3(6), 1899–1902.
20. Maciejewski, A.J., (1995), Non–Integrability of the Planar Oscillations of a Satellite. Acta Astronom- ica, 45, 327–344.
21. Maciejewski, A. J., (2001), Non-integrability of a certain problem of rotational motion of a rigid satel- lite. In Dynamics of Natural and Artificial Celestial Bodies (Edited by Pretka–Ziomek, Richardson), Kluwer Academic Publish, 187–192.
22. Maciejewski, A. J., (1995), The Observer-New method for Numerical Integration of Differential Equa- tions in the Presence of First Integrals. In From Newton to Chaos, (Edited by A. E. Roy and B. A. Steves), Plenum, 503–512.
23. Bogoyavlensky, O.I., (1992), Euler equations on finete–dimensional Lie coalgebras arising in problems of mathematical physics. in Russian, Uspekhi Math. Nauka, 47(1), 107–146; English translation in Russian Math. Surveys 47(1), 117–189.
24. Feng, K., (1987), Lecture Notes in Numerical Methods for P.D.E.´s, Springer–Verlag: New York/Berlin. [25] Li, S.T. and Qin, M., (1995), Lie–Poisson Integration for rigid body dynamics. Computers and Math- ematics with Applications, 30, 105–118.
25. Feng, K., (1984), On difference schemes and symplectic geometry, Beijing Symposium On Differential Equations, (Edited by F. Kang): Beijing.
26. Qin, M. and Zhang, M., (1990), Multi-Stage Symplectic Schemes Of Two Kinds Of Hamiltonian Systems For Wave Equations. Computers and Mathematics with Applications, 19(10), 51–62.
27. McLachlan, R.I. and Quispel, G.R.W., (2002), Splitting Methods, Acta Numerica, 71, 341–434.
28. McLachlan, R.I. and Scovel, C., (1995), Equivariant Constrained Symplectic Integration. J. Nonlinear Science. 5, 233–256.
29. Reich, S., (1994), Momentum Conserving Symplectic Integrator, Physica D, 76, 375–383.
30. Cooper, GJ., (1987), Stability of Runge-Kutta methods for trajectory problems, IMA J. Numer. Anal. 7, 1-13.
31. Suzuki, M. and Umeno, K., (1993), Computer Simulation Studies in Condense Matter Physics IV., Springer, Berlin.
32. Ismail, M.S. and Alamri, S.Z., (2004), Haighly accurate finite difference method for coupled nonlinear Schr¨odinger equation. Journal of Computer and Mathematics. 81(3), 333–351.
33. Ayhan Aydın is an Assist. Prof. Dr. of Mathematics at Atılım Univer- sity, Ankara, Turkey. He was born in Ankara in 1972. He graduated from Middle East Technical University(METU), Department of Mathematics in