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Hamiltonian dynamics of a rigid body in a central gravitational field

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Abstract

This paper concerns the dynamics of a rigid body of finite extent moving under the influence of a central gravitational field. A principal motivation behind this paper is to reveal the hamiltonian structure of the n-body problem for masses of finite extent and to understand the approximation inherent to modeling the system as the motion of point masses. To this end, explicit account is taken of effects arising because of the finite extent of the moving body. In the spirit of Arnold and Smale, exact models of spin-orbit coupling are formulated, with particular attention given to the underlying Lie group framework. Hamiltonian structures associated with such models are carefully constructed and shown to benon-canonical. Special motions, namely relative equilibria, are investigated in detail and the notion of anon-great circle relative equilibrium is introduced. Non-great circle motions cannot arise in the point mass model. In our analysis, a variational characterization of relative equilibria is found to be very useful.

Thereduced hamiltonian formulation introduced in this paper suggests a systematic approach to approximation of the underlying dynamics based on series expansion of the reduced hamiltonian. The latter part of the paper is concerned with rigorous derivations of nonlinear stability results for certain families of relative equilibria. Here Arnold's energy-Casimir method and Lagrange multiplier methods prove useful.

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Authors and Affiliations

  1. Electrical Engineering Department & Systems Research Center, University of Maryland, 20742, College Park, MD

    Li-Sheng Wang & P. S. Krishnaprasad

  2. Mathematics Department, University of Maryland, 20742, College Park, MD

    J. H. Maddocks

Authors
  1. Li-Sheng Wang
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  2. P. S. Krishnaprasad
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  3. J. H. Maddocks
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Additional information

This work was supported in part by the AFOSR University Research Initiative Program under grant AFOSR-87-0073, by AFOSR grant 89-0376, and by the National Science Foundation's Engineering Research Centers Program: NSFD CDR 8803012. The work of P.S. Krishnaprasad was also supported by the Army Research Office through the Mathematical Sciences Institute of Cornell University.

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Wang, LS., Krishnaprasad, P.S. & Maddocks, J.H. Hamiltonian dynamics of a rigid body in a central gravitational field. Celestial Mech Dyn Astr 50, 349–386 (1990). https://doi.org/10.1007/BF02426678

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