Hamiltonian structure of the Vlasov-Einstein system and the problem of stability for spherical relativistic star clusters

Details Hamiltonian structure of the Vlasov-Einstein system and the problem of stability for spherical relativistic star clusters Hamiltonian structure of the Vlasov-Einstein system and the problem of stability for spherical relativistic star clusters

Nov 1992

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1992stin...9322507k&link_type=abstract

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Physics

Boltzmann-Vlasov Equation, Cosmology, Einstein Equations, Gravitation, Hamiltonian Functions, Stability, Degrees Of Freedom, Distribution Functions, Perturbation, Star Clusters

Scientific paper

The Hamiltonian formulation of the Vlasov-Einstein system, which is appropriate for collisionless, self-gravitating systems like clusters of stars that are so dense that gravity must be described by the Einstein equation, is presented. In particular, it is demonstrated explicitly in the context of a 3 + 1 splitting that, for spherically symmetric configurations, the Vlasov-Einstein system can be viewed as a Hamiltonian system, where the dynamics are generated by a noncanonical Poisson bracket, with the Hamiltonian generating the evolution of the distribution function f (a noncanonical variable) being the conserved ADM mass-energy HADM. An explicit expression is derived for the energy (delta2)HADM associated with an arbitrary phase space preserving perturbation of an arbitrary spherical equilibrium, and it is shown that the equilibrium must be linearly stable if (delta2)HADM is positive semi-definite. Insight into the Hamiltonian reformulation is provided by a description of general finite degree of freedom systems.

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