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Are Hamiltonian flows geodesic flows?

Authors: Christopher McCord, Kenneth R. Meyer and Daniel Offin
Journal: Trans. Amer. Math. Soc. 355 (2003), 1237-1250
MSC (2000): Primary 37N05, 34C27, 54H20
Published electronically: 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 $N$-body problem. We show that the flow of the reduced planar $N$-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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Additional Information

Christopher McCord
Affiliation: University of Cincinnati, Cincinnati, Ohio 45221-0025

Kenneth R. Meyer
Affiliation: University of Cincinnati, Cincinnati, Ohio 45221-0025

Daniel Offin
Affiliation: Queen’s University, Kingston, Ontario K7L 4V1, Canada

Keywords: Hamiltonian systems, geodesic flows, Jacobi metric, three--body problem
Received by editor(s): January 2, 2002
Received by editor(s) in revised form: May 10, 2002
Published electronically: October 17, 2002
Additional Notes: This research was partially supported by grants from the Taft Foundation, the NSF and the NSERC
Article copyright: © Copyright 2002 American Mathematical Society

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