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Transactions of the American Mathematical Society

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Mechanical systems with symmetry on homogeneous spaces

Author: Ernesto A. Lacomba
Journal: Trans. Amer. Math. Soc. 185 (1973), 477-491
MSC: Primary 58F05
MathSciNet review: 0331426
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Abstract: The geodesic flow on a homogeneous space with an invariant metric can be naturally considered within the framework of Smale's mechanical systems with symmetry. In this way we have at our disposal the whole method of Smale for studying such systems. We prove that certain sets $ \Sigma ',\Sigma ,\operatorname{Im} ,\sigma $ and Re which play an important role in the global behavior of those systems, have a particularly simple structure in our case, and we also find some geometrical implications about the geodesics. The results obtained are especially powerful for the case of Lie groups, as in the rigid body problem.

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Keywords: Global viewpoint, mechanical systems with symmetry, rigid body problem, kinetic and potential energy, momentum, invariant submanifolds, bifurcation set, relative equilibria, G-invariant metric, algebraic and semialgebraic sets, algebraic manifold, algebraic linear group
Article copyright: © Copyright 1973 American Mathematical Society