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theorems:liouvilles_theorem [2018/05/06 14:04] ida [Intuitive] |
theorems:liouvilles_theorem [2018/07/04 12:35] jakobadmin [Concrete] |
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- | Now, if $\rho$ is constant $\frac{\partial \rho }{\partial t}= 0$, then the left-hand side is $0$ and we get: | + | Now, if $\rho$ is constant $\frac{d \rho }{d t}= 0$, then the left-hand side is $0$ and we get: |
$$\frac{\partial \rho }{\partial t}= -\sum_{i}\left(\frac{\partial \rho}{\partial q_i}\,\dot{q_i}+\frac{\partial\rho}{\partial p_i}\,\dot p_i\right).$$ | $$\frac{\partial \rho }{\partial t}= -\sum_{i}\left(\frac{\partial \rho}{\partial q_i}\,\dot{q_i}+\frac{\partial\rho}{\partial p_i}\,\dot p_i\right).$$ |