Liouville's theorem

Liouville's theorem is an expression of the conservation of volume of phase space:

$\frac{d\varrho }{dt}=\sum _{i=1}^s(\frac{\partial \varrho }{\partial q_i}\dot{q_i}+\frac{\partial \varrho }{\partial p_i}\dot{p_i})=0$

where $\varrho$ is a distribution function $\varrho (p,q)$, p is the generalised momenta and q are the generalised coordinates. With time a volume element can change shape, but phase points neither enter nor leave the volume.