Liouville's theorem

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varrho} is a distribution function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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)