Vlasov and Poisson Equations in the Context of Self‐Gravitating Systems

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.

Keywords:

• Nondissipative physics
• self-gravitating systems
• Vlasov equation
• theory

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

where δ _D is the classical Dirac distribution, and as this distribution is the neutral element of the convolution algebra, one can directly see that Poisson equation can be written as

which is called an inverse Poisson equation.

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

In the case of equation (25) surface terms vanish as the distribution function vanishes at infinity.

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.