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On geometric hydrodynamics and infinite-dimensional magnetic systems

Levin Maier

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 \cite{Arnold66} -- with his formulation of the motion of a charged particle in a magnetic field \cite{ar61}. We introduce the \emph{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.

On geometric hydrodynamics and infinite-dimensional magnetic systems

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 \cite{Arnold66} -- with his formulation of the motion of a charged particle in a magnetic field \cite{ar61}. We introduce the \emph{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.

Paper Structure

This paper contains 18 sections, 15 theorems, 81 equations, 1 table.

Table of Contents

  1. Introduction
  2. Magnetic systems on regular Lie groups
  3. Intermezzo: Magnetic systems
  4. The magnetic Euler--Arnold equation on regular Lie groups
  5. Related results
  6. The one-to-one correspondence
  7. Geodesics on central extensions of regular Lie groups
  8. The correspondence between magnetic geodesics on the Lie group and geodesics on the central extension
  9. Two shallow water equations as magnetic Euler--Arnold equations
  10. The H1 alpha beta--Euler--Arnold equation
  11. Korteweg--de Vries equation as magnetic Euler--Arnold equation
  12. Generalized Camassa--Holm equation as magnetic Euler--Arnold equation
  13. The infinite conductivity equation as a magnetic Euler--Arnold equation
  14. The Global Quasi-Geostrophic Equations as a Magnetic Geodesic Equation
  15. The geometric setting
  16. ...and 3 more sections

Key Result

Lemma 2.3

The magnetic geodesic flow $\varPhi^t_{g,\sigma}$ of $(M,g,\sigma)$ is the Hamiltonian flow induced by the kinetic energy $E\colon TM\to\mathbb{R}$ and the twisted symplectic form where $\lambda$ is the metric pullback of the canonical Liouville $1$-form from $T^*M$ to $TM$, and $\pi_{TM}\colon TM\to M$ is the canonical projection.

Theorems & Definitions (45)

  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3: Gin
  • Definition 2.4
  • Remark 2.5
  • Proposition 2.6
  • Remark 2.7
  • proof
  • Corollary 2.8
  • Theorem 2.9: The magnetic Euler--Arnold equation
  • ...and 35 more