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"Falling cat" connections and the momentum map

Marian Fecko
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Abstract

We consider a standard symplectic dynamics on TM generated by a natural Lagrangian L. The Lagrangian is assumed to be invariant with respect to the action TR_g of a Lie group G lifted from the free and proper action R_g of G on M. It is shown that under these conditions a connection on principal bundle pi: M \rightarrow M/G can be constructed based on the momentum map corresponding to the action TR_g. The horizontal motion is shown to be in physical terms the one with all the momenta corresponding to the symmetry vanishing. A simple explicit formula for the connection form is given. For the special case of the standard action of G = SO(3) on M = R^3 x ... x R^3 corresponding to a rigid rotation of a N-particle system the formula obtained earlier by Guichardet and Shapere/Wilczek is reproduced.

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Marian Fecko. 1997-02-12. "Falling cat" connections and the momentum map. https://doi.org/10.1063/1.531183

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