The-Locally Isotropic Solutions of the Liouville and Poisson Equations

The general solution (~,f) of the Liouville equation for a collisionless system of gravitationally inter- acting particles in which the distribution function f is locally isotropic in momentum space is derived. The distribution function is shown to be a spherical distribution in Chandrasekhar's sense. In the non- stationary case, Poisson's equation is solved and is seen to imply a spatially constant density. The stellar orbits are determined, and a class of anisotropic distribution functions which are also compatible with the previously established potential ~ is constructed. The relation of these solutions to Newtonian cosmology is pointed out