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ISSN 1088-6850(e) ISSN 0002-9947(p)

Are Hamiltonian flows geodesic flows?

Author(s): Christopher McCord; Kenneth R. Meyer; Daniel Offin
Journal: Trans. Amer. Math. Soc. 355 (2003), 1237-1250.
MSC (2000): Primary 37N05, 34C27, 54H20
Posted: October 17, 2002
MathSciNet review: 1938755
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Abstract: When a Hamiltonian system has a ``Kinetic + Potential'' structure, the resulting flow is locally a geodesic flow. But there may be singularities of the geodesic structure; so the local structure does not always imply that the flow is globally a geodesic flow. In order for a flow to be a geodesic flow, the underlying manifold must have the structure of a unit tangent bundle. We develop homological conditions for a manifold to have such a structure.

We apply these criteria to several classical examples: a particle in a potential well, the double spherical pendulum, the Kovalevskaya top, and the -body problem. We show that the flow of the reduced planar -body problem and the reduced spatial 3-body are never geodesic flows except when the angular momentum is zero and the energy is positive.

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