Hamiltonian structure of the Vlasov-Einstein system for generic collisionless systems and the problem of stability

Details Hamiltonian structure of the Vlasov-Einstein system for generic collisionless systems and the problem of stability Hamiltonian structure of the Vlasov-Einstein system for generic collisionless systems and the problem of stability

May 1994

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1994phrvd..49.5115k&link_type=abstract

Physical Review D (Particles, Fields, Gravitation, and Cosmology), Volume 49, Issue 10, 15 May 1994, pp.5115-5125

Physics

3

Canonical Formalism, Lagrangians, And Variational Principles, Kinetic Theory, Relativity And Gravitation, Stellar Dynamics And Kinematics

Scientific paper

Working in the context of an ADM splitting into space plus time, this paper first demonstrates explicitly that the Vlasov-Einstein system, i.e., the collisionless Boltzmann equation of general relativity, is Hamiltonian, and then uses this Hamiltonian character to derive nontrivial criteria for linear and nonlinear stability of time-independent equilibria. Unlike all earlier work on the problem of stability, the formulation provided here is completely general, incorporating no assumptions regarding spatial symmetries or the form of the equilibrium. The fundamental arena of physics is an infinite-dimensional phase space, coordinatized by the distribution function f, the spatial metric hab, and the conjugate momentum Πab. The Hamiltonian formulation entails the identification of a Lie bracket , defined for pairs of functionals F[hab,Πab,f] and G[hab,Πab,f], and a Hamiltonian function H[hab,Πab,f], so chosen that the equations of motion ∂tF= for arbitrary F, with ∂t a coordinate time derivative, are equivalent to the Vlasov-Einstein system. An explicit expression is derived for the most general dynamically accessible perturbation δX≡\{δf,δhab,δΠab\} which satisfies the Hamiltonian and momentum field constraints and the matter constraints associated with conservation of phase, and it is shown that all equilibria are energy extremals with respect to such δX, i.e., δ(1)H[δX]≡0. The sign of the second variation, δ(2)H, also computed, is thus related directly to the problem of linear stability. If δ(2)H>0 for all dynamically accessible perturbations δX, the equilibrium is guaranteed to be linearly stable. The existence of some perturbation δX for which δ(2)H[δX]<0 does not necessarily signal a linear instability. However, one can at least infer that equilibria admitting such perturbations are nonlinearly unstable and/or unstable toward the effects of dissipation.

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