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arXiv · 1806.01755

Averaging, symplectic reduction, and central extensions

Cheng Yang·Boris Khesin
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Abstract

We show that the averaged equation for a one-frequency fast-oscillating Hamiltonian system is the result of symplectic reduction of a certain natural system on the corresponding $S^1$-bundle with respect to the circle action. Furthermore, if the reduced configuration space happens to be a group, then under natural assumptions the averaged system turns out to be the Euler equation on a central extension of that group. This gives a new explanation of the drift, common in averaged system, as a similar shift is typically present in symplectic reductions and central extensions.

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Cheng Yang, Boris Khesin. 2018-06-05. Averaging, symplectic reduction, and central extensions. https://arxiv.org/abs/1806.01755

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