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.

References

1. J. Liouville "Note sur la Théorie de la Variation des constantes arbitraires", Journal de Mathématiques Pures et Appliquées, Sér. I, 3 pp. 342-349 (1838)