Liouville's theorem

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Liouville's theorem is an expression of the conservation of volume of phase space [1]:

\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.

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