Kinematic reduction and the Hamilton-Jacobi equation
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September 2012, 4(3): 239-269. doi: 10.3934/jgm.2012.4.239
Geometry of plasma dynamics II: Lie algebra of Hamiltonian vector fields
Oǧul Esen 1, and Hasan Gümral 1,
Department of Mathematics, Yeditepe University, 34755 Ataşehir, İstanbul, Turkey, Turkey
Received November 2010 Revised March 2012 Published October 2012
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. 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., vertical representative, Poisson-Vlasov equations.
Mathematics Subject Classification: Primary: 70G65, 70H05; Secondary: 22E6.
Citation: Oǧul Esen, Hasan Gümral. Geometry of plasma dynamics II: Lie algebra of Hamiltonian vector fields. Journal of Geometric Mechanics, 2012, 4 (3) : 239-269. doi: 10.3934/jgm.2012.4.239
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Oǧul Esen Hasan Gümral