# Jeans equation for a spherical equilibrium

I am currently studying the Jeans equation for a system in spherical equilibrium. Since the distribution function $f(x,v)$ can be written in depency of energy and angular momentum $f(E, L)$ it seems to follow that most of the velocity moments disappear:

$$\tag{1} \left<\upsilon_r\right> = \left<\upsilon_\vartheta\right> = \left<\upsilon_\varphi\right> = 0$$

and

$$\tag{2} \left<\upsilon_r \upsilon_\vartheta\right> = \left<\upsilon_r \upsilon_\varphi\right> = \left<\upsilon_\vartheta \upsilon_\varphi\right> = 0.$$

Where $f(x,v)$ denotes the number density of particles in phase space and $\left<Q\right> = \frac{m}{\rho(x)} \int Q\,f(x,v) \text{d}^3v$,$\;\;$ $\rho(x) = m \int f(x,v)\text{d}^3 v$.

I don't find it intuitive why these velocity moments are zero. What ist the mathematical way to get these results?