Geodesic Vlasov equations and their integrable moment closures

Darryl D. Holm; Cesare Tronci

doi:10.3934/jgm.2009.1.181

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2009, Volume 1, Issue 2: 181-208. Doi: 10.3934/jgm.2009.1.181

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Geodesic Vlasov equations and their integrable moment closures

Darryl D. Holm^1, and
Cesare Tronci^2,

1.
Department of Mathematics, Imperial College London, London SW7 2AZ
2.
Department of Mathematics, University of Surrey, Guildford GU2 7XH

Received: December 31, 2008

Revised: May 31, 2009

Published: May 2009

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Abstract

Various integrable geodesic flows on Lie groups are shown to arise by taking moments of a geodesic Vlasov equation on the group of canonical transformations. This was already known for both the one- and two-component Camassa-Holm systems [18, 19]. The present paper extends our earlier work to recover another integrable system of ODE's that was recently introduced by Bloch and Iserles [5]. Solutions of the Bloch-Iserles system are found to arise from the Klimontovich solution of the geodesic Vlasov equation. These solutions are shown to form one of the legs of a dual pair of momentum maps. The Lie-Poisson structures for the dynamics of truncated moment hierarchies are also presented in this context.

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Mathematics Subject Classification: Primary: 82C40, 37K05; Secondary: 37N20.

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