Abstract
The Serret-Andoyer transformation is a classical method for reducing the free rigid body dynamics, expressed in Eulerian coordinates, to a 2-dimensional Hamiltonian flow. First, we show that this transformation is the computation, in 3-1-3 Eulerian coordinates, of the symplectic (Marsden-Weinstein) reduction associated with the lifted left-action of SO(3) on T*SO(3)—a generalization and extension of Noether's theorem for Hamiltonian systems with symmetry. In fact, we go on to generalize the Serret-Andoyer transformation to the case of Hamiltonian systems on T*SO(3) with left-invariant, hyperregular Hamiltonian functions. Interpretations of the Serret-Andoyer variables, both as Eulerian coordinates and as canonical coordinates of the co-adjoint orbit, are given. Next, we apply the result obtained to the controlled rigid body with momentum wheels. For the class of Hamiltonian controls that preserve the symmetry on T*SO(3), the closed-loop motion of the main body can again be reduced to canonical form. This simplifies the stability proof for relative equilibria , which then amounts to verifying the classical Lagrange-Dirichlet criterion. Additionally, issues regarding numerical integration of closed-loop dynamics are also discussed. Part of this work has been presented in LumBloch:97a.
Similar content being viewed by others
References
Abesser, H. and Steigenberger, J., “On structure preserving transformations in Hamiltonian control systems,” Zeitschrift f¨ur Analysis und ihre Anwendungen, vol. 10, no. 3, pp. 319–333, 1991.
Abraham, R. and Marsden, J. E., Foundations of Mechanics, Addison-Wesley, 2nd edition, 1985.
Andoyer, H., Cours de Mxécanique Céleste, Paris: Gauthier-Villars, 1923.
Arnold, V. I., “Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits,” Annales de l'Institut Fourier (Grenoble), vol. 16, pp. 319–361, 1966.
Arnold, V. I., Mathematical Methods of Classical Mechanics, in Graduate Texts in Mathematics, New York: Springer-Verlag, 2nd edition, 1988.
Austin, M., Krishnaprasad, P. S. and Wang, L.-S., “Almost Poisson integration of rigid body systems,” J. Computational Physics, vol. 107, pp. 105–117, 1993.
Bloch, A. M., Krishnaprasad, P. S., Marsden, J. E. and Sánchez de Alvarez, G., “Stabilization of rigid body dynamics by internal and external torques,” Automatica, vol. 28, pp. 745–756, 1992.
Bloch, A. M., Marsden, J. E. and Sánchez de Alvarez, G., “Feedback stabilization of relative equilibria for mechanical systems with symmetry,” in Current and Future Directions in Applied Mathematics, pp. 43–64, Notre Dame, IN, 1997. Boston: Birkhauser.
Channell, P. J. and Scovel, C., “Symplectic integration of Hamiltonian systems,” Nonlinearity, vol. 3, pp. 231–259, 1990.
Crouch, P. E. and Grossman, R., “Numerical integration of ordinary differential equations on manifolds,” Journal of Nonlinear Science, vol. 3, pp. 1–33, 1993.
Deprit, A. and Elipe, A., “Complete reduction of the Euler-Poinsot problem,” J. Astronautical Sciences, vol. 41, pp. 603–628, 1993.
Feng, K. and Ge, Z., “On the approximation of linear Hamiltonian systems,” J. Computational Mathematics, vol. 6, pp. 88–97, 1988.
Ge, Z. and Marsden, J. E., “Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators,” Physics Letters A, vol. 133, pp. 134–139, 1988.
Greenwood, D. T., Principles of Dynamics, Englewood Cliffs, N.J.: Prentice-Hall, 2nd edition, 1988.
Hildebrand, F. B., Finite-Difference Equations and Simulations, Englewood Cliffs, N.J.: Prentice-Hall, Inc., 1968.
Lum, K.-Y. and Bloch, A. M., “A Serret-Andoyer transformation analysis of the controlled rigid body,” in Proc. 36th IEEE Conference on Decision an Control, San Diego, December 10–12, 1997.
Marsden, J. E., Lectures on Mechanics, London Mathematical Society Lecture Note Series 174. U.K.: Cambridge University Press, 1992.
Marsden, J. E., Patrick, G.W. and Schadwick, W. F. (eds.), Integration Algorithms and Classical Mechanics, in Fields Institute Communications, Providence, R.I.: American Mathematical Society, volume 10, 1996.
Marsden, J. E. and Ratiu, T. S., Introduction to Mechanics and Symmetry, Text in Applied Mathematics 17. New York: Springer-Verlag, 1994.
Marsden, J. E. and Weinstein, A., “Reduction of symplectic manifolds with symmetry,” Reports on Mathematical Physics, vol. 5, pp. 121–130, 1974.
Marsden, J. E. and Wendlandt, J.,M., “Mechanical systems with symmetry, variational principles, and integration algorithms,” in Current and future directions in applied mathematics, pp. 219–261, Notre Dame, IN, 1997. Boston: Birkhauser.
Reich, S., “Symplectic integrators for systems of rigid bodies,” in Marsden, J. E., Patrick, G.W. and Schadwick, W. F. (eds.), Integration Algorithms and Classical Mechanics, Providence, RI: American Mathematical Society, 1996, pp. 181–191.
Serret, J. A., “Mémoire sur l'emploi de la méthode de la variation des arbitraires dans la théorie des mouvements de rotation,” Mémoires de l'académie des sciences de Paris, vol. 35, pp. 585–616, 1866.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Lum, KY., Bloch, A.M. Generalized Serret-Andoyer Transformation and Applications for the Controlled Rigid Body. Dynamics and Control 9, 39–66 (1999). https://doi.org/10.1023/A:1008342708491
Issue Date:
DOI: https://doi.org/10.1023/A:1008342708491