Liouville's theorem

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

${\frac {d\varrho }{dt}}=\sum _{i=1}^{s}\left({\frac {\partial \varrho }{\partial q_{i}}}{\dot {q_{i}}}+{\frac {\partial \varrho }{\partial p_{i}}}{\dot {p_{i}}}\right)=0$

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.