Hamiltonian dynamics of a rigid body in a central gravitational field |
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| Authors: | Li-Sheng Wang P S Krishnaprasad J H Maddocks |
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| Institution: | (1) Electrical Engineering Department & Systems Research Center, University of Maryland, 20742 College Park, MD;(2) Mathematics Department, University of Maryland, 20742 College Park, MD |
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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.
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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| Keywords: | Hamiltonian mechanics relative equilibrium rigid body dynamics |
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