Are Hamiltonian flows geodesic flows?

McCord, Christopher; Meyer, Kenneth; Offin, Daniel

doi:10.1090/S0002-9947-02-03167-7

Are Hamiltonian flows geodesic flows?
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by Christopher McCord, Kenneth R. Meyer and Daniel Offin

Trans. Amer. Math. Soc. 355 (2003), 1237-1250

DOI: https://doi.org/10.1090/S0002-9947-02-03167-7

Published electronically: October 17, 2002

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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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