Hamiltonian structure of the Vlasov-Maxwell system in a curved background spacetime.

Abstract

Working in the context of an ADM splitting into space plus time, this paper identifies a Hamiltonian formulation, i.e., cosymplectic structure, for the Vlasov-Maxwell system, as formulated in a fixed, albeit curved, background spacetime. The fundamental arena of physics is an infinite-dimensional phase space, coordinatized by the distribution function f, the spatial vector potential ${\mathit{A}}_{\mathit{i}}$, and the conjugate momentum ${\mathrm{\ensuremath{\Pi}}}^{\mathit{i}}$. This Hamiltonian formulation entails the identification of (i) Lie brackets 〈F,G〉, defined for functionals F[${\mathit{A}}_{\mathit{i}}$,${\mathrm{\ensuremath{\Pi}}}^{\mathit{i}}$,f] and G[${\mathit{A}}_{\mathit{i}}$,${\mathrm{\ensuremath{\Pi}}}^{\mathit{i}}$,f], and (ii) a Hamiltonian function H[${\mathit{A}}_{\mathit{i}}$,${\mathrm{\ensuremath{\Pi}}}^{\mathit{i}}$,f], so chosen that the equations of motion take the form ${\mathrm{\ensuremath{\partial}}}_{\mathit{t}}$F=〈H,F〉, with ${\mathrm{\ensuremath{\partial}}}_{\mathit{t}}$ a coordinate time derivative. This formulation is used to address the problem of stability of equilibria corresponding to time-independent electromagnetic fields in a spacetime admitting a timelike Killing field. An explicit expression is derived for the most general dynamically accessible perturbation \ensuremath{\delta}X\ensuremath{\equiv}(\ensuremath{\delta}${\mathit{A}}_{\mathit{i}}$,\ensuremath{\delta}${\mathrm{\ensuremath{\Pi}}}^{\mathit{i}}$,\ensuremath{\delta}f), and it is shown that all equilibria are energy extremals with respect to such \ensuremath{\delta}X, i.e., ${\mathrm{\ensuremath{\delta}}}^{(1)}$H[\ensuremath{\delta}X]\ensuremath{\equiv}0. The sign of the second variation ${\mathrm{\ensuremath{\delta}}}^{(2)}$H, also computed, is thus directly related to the problem of linear stability: If ${\mathrm{\ensuremath{\delta}}}^{(2)}$H[\ensuremath{\delta}X]>0 for all dynamically accessible perturbations, the equilibrium is guaranteed to be linearly stable. The existence of some perturbation \ensuremath{\delta}X for which ${\mathrm{\ensuremath{\delta}}}^{(2)}$H[\ensuremath{\delta}X]0 does not guarantee a linear instability. However, one does at least anticipate that equilibria admitting such perturbations will be nonlinearly unstable and/or unstable towards the effects of dissipation.