On geometric hydrodynamics and infinite dimensional magnetic systems

2025/05/31 by Maier, Levin · 3 citations
#Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2506.00544

Abstract

In this article, we combine V. Arnold's celebrated approach via the Euler-Arnold equation -- describing the geodesic flow on a Lie group equipped with a right-invariant metric~\citeArnold66 -- with his formulation of the motion of a charged particle in a magnetic field~\citear61. We introduce the magnetic Euler-Arnold equation, which is the Eulerian form of the magnetic geodesic flow for an infinite-dimensional magnetic system on a Lie group endowed with a right-invariant metric and a right-invariant closed two-form serving as the magnetic field. As an illustration, we demonstrate that the Korteweg--de Vries equation, the generalized Camassa-Holm equation, the infinite conductivity equation, and the global quasi-geostrophic equations can all be interpreted as magnetic Euler-Arnold equations. In particular, we obtain both local and global well-posedness results for the magnetic Euler-Arnold equation associated with the global quasi-geostrophic equations.

Cited by

• On Mañé's Critical Value for Tonelli Lagrangians on Half Lie-Groups
• The Hopf--Rinow Theorem and Mañé's Critical Value for Magnetic Geodesics on Half Lie-Groups
• On the contact type conjecture for exact magnetic systems

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