Vlasov moment flows geodesics on the Jacobi group

By using the moment algebra of the Vlasov kinetic equation, we characterize the integrable Bloch-Iserles system on symmetric matrices [Bloch, A. M., Brînzǎnescu, V., Iserles, A., Marsden, J. E., Ratiu, T. S., "A class of integrable flows on the space of symmetric matrices," Commun. Math. Phys.290, 399-435 (2009)]10.1007/s00220-009-0849-6 as a geodesic flow on the Jacobi group .We analyze the corresponding Lie-Poisson structure by presenting a momentum map, which both untangles the bracket structure produces particle-type solutions that are inherited from the Vlasov-like interpretation. Moreover, we show how the Vlasov moments associated to Bloch-Iserles dynamics correspond to particular subgroup inclusions into a group central extension (first discovered by Ismagilov, Losik, Michor ["A 2-cocycle on a group of symplectomorphisms," Mosc. Math. J.6, 307-315 (2006)]), which in turn underlies Vlasov kinetic theory. In the most general case of Bloch-Iserles dynamics, a generalization of the Jacobi group also emerges naturally.