Abstract
We present Vlasov's equation and its association with Poisson's equation in the context of modelling self‐gravitating systems such as globular clusters or galaxies. We first review the classical hypotheses of the model. We continue with a presentation of the Hamilton‐Morrison structure of Vlasov's equation to study the equilibrium and the stability of self‐gravitating systems. Finally, we present some preliminary results concerning some properties of the time‐dependent solutions of the Vlasov‐Poisson system.
Notes
1In particular, this means that all particles have the same mass m.
2Poisson equation writes Δψ (q, t)=4π Gρ (q, t) where the mass density is directly related to the distribution function by ρ (q, t)=m∫ f (q, p′, t) d p′. As one can verify in 3D that
3This is actually an integration by parts. One can directly show indeed that for functions that decay sufficiently rapidly at |q| and |p| tend to ∞ we have
4By equilibrium, we mean here a stationary solution of the Vlasov‐Poisson system.
5This proposition is not reciprocal.
6Classically, the characteristic function is defined to be unity on Ω, and is zero elsewhere.