Averaging, symplectic reduction, and central extensions

2018/06/05 by Cheng Yang, Yang, Cheng, Boris Khesin +1 · 1 citation
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Geometry and complex manifolds #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.1806.01755

openalex publication_date 2018/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 S¹-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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