Liouville Theorem analogue in generalized velocities? The Liouville Theorem concerns dynamics in phase space: does an analogue exist in configuration space, and, if not, could you give a motivation / proof why?
 A: Here is a direct counterexample to complete Qmechanic's answer.
 Take $L= q \dot{q}^2/2$ for $q>0$. As a consequence
$$p = q\dot{q}$$
and so
$$dp = q d\dot{q} + \dot{q} dq\:.$$
Therefore, the canonical volume in terms of Lagrangian variables is
$$dp\wedge dq = q d\dot{q} \wedge dq\:.$$
Since the left-hand side is preserved by solutions of the equation of motion and $q=q(t)$ is not constant in time along these solutions, it is not possible that 
$$ d\dot{q} \wedge dq$$
is also constant along the motion of the system. So the apparantly natural volume constructed out of Lagrangian variables is not constant in time on the motion of the system, differently from the canonical volume.
A: Unlike in the cotangent bundle phase space $T^{\ast}M$, there is no generally valid Liouville Theorem (where time evolution is divergencefree) in the base configuration space $M$ nor in its tangent bundle $TM$. For starters, the notion of divergence needs a notion of volume, and neither $M$ nor $TM$ is generically equipped with a canonical volume form.
