The Hamiltonian structure of the Maxwell-Vlasov equations. (English) Zbl 1194.35463
Summary: P.J. Morrison [The Maxwell-Vlasov equations as a continuous Hamiltonian system. Phys. Lett. 80A, 383–386 (1980)] has observed that the Maxwell-Vlasov and Poisson-Vlasov equations for a collisionless plasma can be written in Hamiltonian form relative to a certain Poisson bracket. We derive another Poisson structure for these equations by using general methods of symplectic geometry. The main ingredients in our construction are the symplectic structure on the co-adjoint orbits for the group of canonical transformations, and the symplectic structure for the phase space of the electromagnetic field regarded as a gauge theory. Our Poisson bracket satisfies the Jacobi identity, whereas Morrison’s does not [A. Weinstein and P.J. Morrison, Comments on: The Maxwell-Vlasov equations as a continuous Hamiltonian system. Phys. Lett. 86A, 235–236 (1981)]. Our construction also shows where canonical variables can be found and can be applied to the Yang-Mills-Vlasov equations and to electromagnetic fluid dynamics.
MSC:
| 35Q83 | Vlasov equations |
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