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Weinstein and P. Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Dongming Wei. Pierre-Emmanuel Jabin. A review of the mean field limits for Vlasov equations. Darryl D. Holm , Cesare Tronci. Geodesic Vlasov equations and their integrable moment closures. Journal of Geometric Mechanics , , 1 2 : The Tulczyjew triple in mechanics on a Lie group. Journal of Geometric Mechanics , , 8 4 : Katherine Zhiyuan Zhang.
Focusing solutions of the Vlasov-Poisson system. Gradient flow structure for McKean-Vlasov equations on discrete spaces. The Vlasov-Navier-Stokes equations as a mean field limit.
Robert T. Glassey , Walter A. Perturbation of essential spectra of evolution operators and the Vlasov-Poisson-Boltzmann system. Hyung Ju Hwang , Juhi Jang. Combined quasineutral and inviscid limit of the Vlasov-Poisson-Fokker-Planck system. Carrillo , Chi-Wang Shu. Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system. Time evolution of a Vlasov-Poisson plasma with magnetic confinement. Gang Li , Xianwen Zhang. A Vlasov-Poisson plasma of infinite mass with a point charge. Laurent Bernis , Laurent Desvillettes. Propagation of singularities for classical solutions of the Vlasov-Poisson-Boltzmann equation.
Jack Schaeffer. Global existence for the Vlasov-Poisson system with steady spatial asymptotic behavior. Lagrangian solutions to the Vlasov-Poisson system with a point charge. Yemin Chen.
Methods of Differential Geometry in Analytical Mechanics, Volume - 1st Edition
Smoothness of classical solutions to the Vlasov-Poisson-Landau system. The two dimensional Vlasov-Poisson system with steady spatial asymptotics. Meixia Xiao , Xianwen Zhang. On global solutions to the Vlasov-Poisson system with radiation damping.
American Institute of Mathematical Sciences. Previous Article Invariant sets forced by symmetry. We introduce natural differential geometric structures underlying the Poisson-Vlasov equations in momentum variables. First, we decompose the space of all vector fields over particle phase space into a semi-direct product algebra of Hamiltonian vector fields and its complement. The latter is related to dual space of the Lie algebra. We identify generators of homotheties as dynamically irrelevant vector fields in the complement. Lie algebra of Hamiltonian vector fields is isomorphic to the space of all Lagrangian submanifolds with respect to Tulczyjew symplectic structure.
This is obtained as tangent space at the identity of the group of canonical diffeomorphisms represented as space of sections of a trivial bundle. We obtain the momentum-Vlasov equations as vertical equivalence, or representative, of complete cotangent lift of Hamiltonian vector field generating particle motion.
Vertical representatives can be described by holonomic lift from a Whitney product to a Tulczyjew symplectic space. We show that vertical representatives of complete cotangent lifts form an integrable subbundle of this Tulczyjew space. We exhibit dynamical relations between Lie algebras of Hamiltonian vector fields and of contact vector fields, in particular; infinitesimal quantomorphisms on quantization bundle.
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Gauge symmetries of particle motion are extended to tensorial objects including complete lift of particle motion. Poisson equation is then obtained as zero value of momentum map for the Hamiltonian action of gauge symmetries for kinematical description. Keywords: complete lifts , momentum-Vlasov equations , Tulczyjew triple.
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