Hamiltonian structure of collisionless Newtonian cosmologies and the problem of stability

Details Hamiltonian structure of collisionless Newtonian cosmologies and the problem of stability Hamiltonian structure of collisionless Newtonian cosmologies and the problem of stability

Apr 1993

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1993phrvd..47.3229k&link_type=abstract

Physical Review D (Particles, Fields, Gravitation, and Cosmology), Volume 47, Issue 8, 15 April 1993, pp.3229-3234

Physics

2

Kinetic Theory

Scientific paper

This paper exhibits a cosymplectic structure for the collisionless Boltzmann (i.e., gravitational Vlasov-Poisson) equation in the context of an expanding Newtonian cosmology, working in the (noninertial) average comoving frame. This formulation entails the identification of an appropriate Lie bracket and a time-dependent Hamiltonian H(t), in terms of which an evolution governed by the collisionless Boltzmann equation involves a generalized canonical transformation. It is observed that all equilibrium solutions f0 corresponding to homogeneous distributions of matter are energy extrema with respect to dynamically accessible perturbations that preserve all the constraints associated with conservation of phase, i.e., δ(1)H≡0. A simple expression is derived for the second variation δ(2)H associated with an arbitrary dynamically accessible perturbation, and it is shown that the time derivative dδ(2)H/dt is intrinsically negative. It follows that any such perturbation with δ(2)H<0 necessarily triggers a linear instability. This provides a simple proof of the linear instability of all equilibria towards perturbations of characteristic wavelength longer than the Jeans length λJ, and of the linear instability of equilibria exhibiting population inversions towards shorter-wavelength perturbations as well.

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