Hamiltonian Formalism of Two-Dimensional Vlasov Kinetic Equation

Abstract

In this paper the two-dimensional Benney system describing long wave propagation of a finite depth fluid motion and the multi-dimensional Russo--Smereka kinetic equation describing a bubbly flow are considered. The Hamiltonian approach established by J. Gibbons for one-dimensional Vlasov kinetic equation is extended to a multi-dimensional case. A local Hamiltonian structure associated with the hydrodynamic lattice of moments derived by D.J. Benney is constructed. A relationship between this hydrodynamic lattice of moments and the two-dimensional Vlasov kinetic equation is found. In the two-dimensional case a Hamiltonian hydrodynamic lattice for the Russo--Smereka kinetic model is constructed. Simple hydrodynamic reductions are presented.

keywords: collisionless kinetic equation, Hamiltonian structure, hydrodynamic lattice, hydrodynamic reduction

References in corpus (3)

• Geometry of Vlasov kinetic moments: a bosonic Fock space for the symmetric Schouten bracket
• The Hamiltonian approach in classification and integrability of hydrodynamic chains
• Differential Geometry of Hydrodynamic Vlasov Equations

Cited by in corpus (1)

• A Poisson map from kinetic theory to hydrodnamics with non-constant entropy