Averaging, symplectic reduction, and central extensions

TL;DR

This paper links the process of averaging in fast-oscillating Hamiltonian systems to symplectic reduction and central extensions, providing a geometric explanation for the drift observed in averaged systems.

Contribution

It introduces a novel geometric perspective connecting averaging, symplectic reduction, and central extensions in Hamiltonian systems.

Findings

Averaged equations arise from symplectic reduction on $S^1$-bundles.

When the reduced space is a group, the averaged system is an Euler equation on a central extension.

Provides a geometric explanation for the drift in averaged systems.

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.