# American Institute of Mathematical Sciences

June  2009, 1(2): 181-208. doi: 10.3934/jgm.2009.1.181

## Geodesic Vlasov equations and their integrable moment closures

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

Received  January 2009 Revised  June 2009 Published  July 2009

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.
Citation: Darryl D. Holm, Cesare Tronci. Geodesic Vlasov equations and their integrable moment closures. Journal of Geometric Mechanics, 2009, 1 (2) : 181-208. doi: 10.3934/jgm.2009.1.181
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