A second look at the Kurth solution in galactic dynamics

The Kurth solution is a particular non-isotropic steady state solution to the gravitational Vlasov-Poisson system. It has the property that by means of a suitable time-dependent transformation it can be turned into a family of time-dependent solutions. Therefore, for a general steady state $ Q(x, v) = \tilde{Q}(e_Q, \beta) $, depending upon the particle energy $ e_Q $ and $ \beta = \ell^2 = |x\wedge v|^2 $, the question arises if solutions $ f $ could be generated that are of the form

$ f(t) = \tilde{Q}\Big(e_Q(R(t), P(t), B(t)), B(t)\Big) $

for suitable functions $ R $, $ P $ and $ B $, all depending on $ (t, r, p_r, \beta) $ for $ r = |x| $ and $ p_r = \frac{x\cdot v}{|x|} $. We are going to show that, under some mild assumptions, basically if $ R $ and $ P $ are independent of $ \beta $, and if $ B = \beta $ is constant, then $ Q $ already has to be the Kurth solution.

This paper is dedicated to the memory of Professor Robert Glassey.