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\centerline{\bf Solution for assignment 20 set~5} \medskip

\centerline{\bf Liouville's Theorem}
\bigskip

\noindent We consider motion of point particles with $n$ degrees
of freedom in {\bf phase space}, which is described by a Hamiltonian 
$$H(q_1,\dots,q_n;p_1,\dots,p_n)\,.$$
Let $\rho(q_1,\dots,q_n;p_1,\dots,p_n;t)$ be the density in phase
space and the velocity of the density element is the vector
$$   \vec{v}\ =\ (q_1,\dots,q_n;p_1,\dots,p_n)\,. $$ 
The gradient is now also defined in phase space $(\hat{q}_i$ and
$\hat{p}_i$ are unit vectors):
$$\nabla\ =\ \sum_{i=1}^n \left(\hat{q}_i\frac{\partial~}
  {\partial q_i}+\hat{p}_i\frac{\partial~}{\partial p_i}\right)\,.$$
The continuity equation reads
$$ 0\ =\ \frac{\partial\rho}{\partial t}+\nabla\cdot\vec{j}\ =\
   \frac{\partial\rho}{\partial t}+\nabla\cdot\left(\vec{v}\rho\right)$$
as $\vec{j}=\vec{v}\rho$ holds. Therefore,
$$ 0\ =\ \frac{\partial\rho}{\partial t} + \rho\,\nabla\cdot\vec{v}
      +  \vec{v}\cdot\nabla\,\rho \ =\ \frac{\partial\rho}{\partial t} 
      + \rho\,\sum_{i=1}^n \left(\frac{\partial\dot{q}_i}{\partial q_i}
      + \frac{\partial\dot{p}_i}{\partial p_i}\right) + \sum_{i=1}^n 
        \left( \frac{\partial\rho}{\partial q_i}\,\dot{q}_i + 
         \frac{\partial\rho}{\partial p_i}\,\dot{p}_i\right)\,. $$
Using Hamilton's equations we have
$$ \frac{\partial\dot{q}_i}{\partial q_i}\ =\ +
   \frac{\partial~}{\partial q_i}\,\frac{\partial H}{\partial p_i}
   ~~{\rm and}~~ \frac{\partial\dot{o}_i}{\partial p_i}\ =\ -
   \frac{\partial~}{\partial p_i}\,\frac{\partial H}{\partial q_i}\,.$$
Interchanging the derivative these terms cancel one another and we
are left with Liouville's theorem:
$$ 0\ =\ \frac{\partial\rho}{\partial t} + \sum_{i=1}^n \left( 
         \frac{\partial\rho}{\partial q_i}\,\dot{q}_i + 
         \frac{\partial\rho}{\partial p_i}\,\dot{p}_i\right)\
      =\ \frac{d\rho}{dt}\,. $$
This is the motion of an incompressible fluid, but in phase
space instead of coordinate space.
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