Vlasov moment flows and geodesics on the Jacobi group

Abstract

By using the moment algebra of the Vlasov kinetic equation, we characterize the integrable Bloch-Iserles system on symmetric matrices (arXiv:math-ph/0512093) 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 and 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 in arXiv:math/0410100), 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.

45 pages

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Cited by in corpus (7)

• Koopman wavefunctions and classical-quantum correlation dynamics
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• Geometric dynamics on the automorphism group of principal bundles: geodesic flows, dual pairs and chromomorphism groups
• Geometry and Dynamics of Gaussian Wave Packets and their Wigner Transforms
• Hamiltonian approach to Ehrenfest expectation values and Gaussian quantum states
• Hamiltonian reductions of the one-dimensional Vlasov equation using phase-space moments