# Is it possible to obtain an analogous of Liouville's Theorem in terms of the Configuration Space?

Is it possible to obtain an analogous of Liouville's Theorem in terms of the Configuration Space (or the tangent space thereof)? If not, why is this result obtainable only in the Hamiltonian formulation? According to several books some of the results in classical mechanics are only possible, or at least obvious, due to the Hamiltonian and Phase Space properties, but the reason has never been clear to me. What are the fundamental differences between the Configuration and Phase Spaces, besides the obvious distinction between the set of variables used by them.

No, not in general. For starter, unlike the cotangent space $T^{\ast}M$, which has a canonical volume form $$\Omega~=~\frac{1}{n!}\omega^{\wedge n},\tag{1}$$ where $$\omega~=~\sum_{i=1}^n \mathrm{d}p_i\wedge \mathrm{d}q^i \tag{2}$$ is the canonical symplectic 2-form; there is no natural choice of volume form in the tangent space $TM$ or in the base manifold $M$.