Celestial Mechanics and Dynamical Astronomy

, Volume 98, Issue 2, pp 121–144 | Cite as

Lie group variational integrators for the full body problem in orbital mechanics

  • Taeyoung Lee
  • Melvin Leok
  • N. Harris McClamroch
Original Article


Equations of motion, referred to as full body models, are developed to describe the dynamics of rigid bodies acting under their mutual gravitational potential. Continuous equations of motion and discrete equations of motion are derived using Hamilton’s principle. These equations are expressed in an inertial frame and in relative coordinates. The discrete equations of motion, referred to as a Lie group variational integrator, provide a geometrically exact and numerically efficient computational method for simulating full body dynamics in orbital mechanics; they are symplectic and momentum preserving, and they exhibit good energy behavior for exponentially long time periods. They are also efficient in only requiring a single evaluation of the gravity forces and moments per time step. The Lie group variational integrator also preserves the group structure without the use of local charts, reprojection, or constraints. Computational results are given for the dynamics of two rigid dumbbell bodies acting under their mutual gravity; these computational results demonstrate the superiority of the Lie group variational integrator compared with integrators that are not symplectic or do not preserve the Lie group structure.


Symplectic integrator Variational integrator Lie group method Full rigid body problem 


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  1. 1.
    Breiter S., Melendo B., Bartczak P. and Wytrzyszczak I. (2005a). Synchronous motion in the Kinoshita problem. Astron. Astrophys. 437: 753–764 CrossRefADSGoogle Scholar
  2. 2.
    Breiter S., Nesvorný D. and Vokrouhlický D. (2005b). Efficient Lie-Poisson integrator for secular spin dynamics of rigid bodies. Astron. J. 130: 1267–1277 CrossRefADSGoogle Scholar
  3. 3.
    Cendra, H., Marsden, J.E., Ratiu, T.S.: Lagrangian reduction by stages. Mem. Am. Math. Soc. 152 (2001)Google Scholar
  4. 4.
    Fahnestock, E., Lee, T., Leok, M., McClamroch, N.H., Scheeres, D.J.: Polyhedral potential and variational integrator computation of the full two body problem. In: AIAA/AAS Astrodynamics Specialist Meeting. AIAA-2006-6289 (2006)Google Scholar
  5. 5.
    Ge Z. and Marsden J.E. (1988). Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators. Phys. Lett. A 133(3): 134–139 CrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Hairer E. (1994). Backward analysis of numerical integrators and symplectic methods. Ann. Numer. Math. 1: 107–132 scientific computation and differential equations (Auckland, 1993).zbMATHMathSciNetGoogle Scholar
  7. 7.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration, 2nd edn. Springer (2006)Google Scholar
  8. 8.
    Iserles A., Munthe-Kaas H.Z., Nørsett S.P. and Zanna A. (2000). Lie-group methods. Acta Numerica 9: 215–365 CrossRefGoogle Scholar
  9. 9.
    Jalnapurkar S.M., Leok M., Marsden J.E. and West M. (2006). Discrete Routh reduction. J. Physica A 39: 5521–5544 zbMATHCrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Kane C., Marsden J.E. and Ortiz M. (1999). Symplectic-energy-momentum preserving variational integrators. J. Math. Phys. 40(7): 3353–3371 zbMATHCrossRefADSMathSciNetGoogle Scholar
  11. 11.
    Krysl P. (2005). Explicit momentum-conserving integrator for dynamics of rigid bodies approximating the midpoint Lie algorithm. Int. J. Numer. Methods Eng. 63: 2171–2193 zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Lee, T., Leok, M., McClamroch, N.H.: A Lie group variational integrator for the attitude dynamics of a rigid body with applications to the 3D pendulum. In: Proceedings of the IEEE Conference on Control Applications, pp. 962–967 (2005)Google Scholar
  13. 13.
    Lee, T., Leok, M., McClamroch, N.H.: Lie group variational integrators for the full body problem. Comput. Methods App. Mech. Eng. in press (2007)Google Scholar
  14. 14.
    Leok, M.: Foundations of Computational Geometric Mechanics. Ph.D. thesis, California Instittute of Technology (2004)Google Scholar
  15. 15.
    Maciejewski A.J. (1995). Reduction, relative equilibria and potential in the two rigid bodies problem. Celestial Mech. Dyn. Astr. 63: 1–28 zbMATHCrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Margot J.L., Nolan M.C., Benner L.A.M., Ostro S.J., Jurgens R.F., Giorgini J.D., Slade M.A. and Campbell D.B. (2002). Binary asteroids in the Near-Earth object population. Science 296: 1445–1448 CrossRefADSGoogle Scholar
  17. 17.
    Marsden J.E., Pekarsky S. and Shkoller S. (1999). Discrete Euler–Poincaré and Lie–Poisson equations. Nonlinearity 12: 1647–1662 zbMATHCrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Marsden J.E., Pekarsky S. and Shkoller S. (2000). Symmetry reduction of discrete Lagrangian mechanics on Lie groups. J. Geom. Phys. 36: 139–150 CrossRefMathSciNetGoogle Scholar
  19. 19.
    Marsden J.E. and West M. (2001). Discrete mechanics and variational integrators. Acta Numerica 10: 357–514 zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Moser J. and Veselov A.P. (1991). Discrete versions of some classical integrable systems and factorization of matrix polynomials. Commun. Math. Phys. 139: 217–243 zbMATHCrossRefADSMathSciNetGoogle Scholar
  21. 21.
    Scheeres D.J. (2002). Stability in the Full Two-Body problem. Celestial Mech. Dyn. Astr. 83: 155–169 zbMATHCrossRefADSMathSciNetGoogle Scholar
  22. 22.
    Simo J.C., Tarnow N. and Wong K.K. (1992). Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics. Comput. Methods Appl. Mech. Eng. 100: 63–116 zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Touma J. and Wisdom J. (1994). Lie-Poisson integrator for rigid body dynamics in the solar system. Astron. J. 107: 1189–1202 CrossRefADSGoogle Scholar
  24. 24.
    Wendlandt J.M. and Marsden J.E. (1997). Mechanical integrator derived from a discrete variational principle. Physica D 106: 223–246 zbMATHCrossRefADSMathSciNetGoogle Scholar
  25. 25.
    Werner R.A. and Scheeres D.J. (2005). Mutual potential of homogenous polyhedra. Celest. Mech. and Dyn. Astr. 91: 337–349 zbMATHCrossRefADSMathSciNetGoogle Scholar
  26. 26.
    Yoshida H. (1990). Construction of high order symplectic integrators. Phys. Lett. A 150: 262–268CrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  • Taeyoung Lee
    • 1
  • Melvin Leok
    • 2
  • N. Harris McClamroch
    • 1
  1. 1.Department of Aerospace EngineeringThe University of MichiganAnn ArborUSA
  2. 2.Department of MathematicsPurdue UniversityWest LafayetteUSA

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