Two different definitions of a Liouville measure

Question

Ok, I'm currently confused because of two different definitions for the Liouville measure associated to a smooth manifold $M$ of dimension $n$. These are:

a) The measure $\mu$ on the cotangent bundle $T^*M$ induced by the volume form $(d\alpha)^n$ where $\alpha$ is the canonical 1-form on $T^*M$ and $\omega:=d\alpha$ makes $T^*M$ a symplectic manifold.

This $\mu$ is invariant under any Hamiltonian flow (that is, any flow that integrates a vector field $X_H$ defined for some smooth $H:T^*M\to \mathbb{R}$ by $\theta(X_H)=\omega(\theta,dH)$ for all $\theta\in T^*(T^*M)$ ).

b) The measure $\nu$ that may be defined, once $M$ is endowed with a Riemannian metric $g$, on $SM$, the unit tangent bundle. This measure is given locally by the product of the Riemannian volume on $M$ (i.e. $\text{det}(g_{ij})dx_1\wedge\dots\wedge dx_n$) and the usual Lebesgue measure on the unit sphere.

This $\nu$ is invariant under the geodesic flow on $SM$.

Now, what is, if any, the relationship between these measures? I imagine that once $M$ is endowed with $g$ we can identify $T^*M$ with $TM$ ($q,p\to q,v$ iff $p(v')=g(v,v')$ for al $v'\in T_qM$). Will then the restriction of $\mu$ to the unit cotangent bundle (assuming this restriction to a submanifold of smaller dimension is well defined) correspond to $\nu$? It's the only reason I could imagine for giving the same name to two different measures defined on two different spaces ($T^*M$ and $SM$, respectively), but I could not find such a statement on any reference book.

Answer 1

I can't say for sure that it is the reason why the two notions are named the same, since they are not equivalent, but you have identified a relation between the two. In a way, the second notion is a special case of the first.

Let $(M,g)$ be a Riemannian manifold. As such, there exists a monomorphism of vector bundles

$$\flat : TM \to T^*M : v \mapsto v^{\flat} = v \, \lrcorner \, g \; .$$

Since both bundles have the same (finite) rank, it is also an epimorphism and so an isomorphism : there exists an inverse morphism

$$\sharp : TM \to T^*M : \lambda \mapsto \lambda^{\sharp} \; .$$

We can define a bundle metric $g^*$ on $T^*M$ as follows : for $\alpha, \beta \in T^*_mM$, let $g^*(\alpha, \beta) = g(\alpha^{\sharp}, \beta^{\sharp})$. This means that $\flat$ and $\sharp$ are "bundle isometries" : $(TM, g)$ is somewhat the same as $(T^*M, g^*)$. As such, the map $\flat$ sends the Liouville measure of and the geodesic flow on $(TM, g)$ to what we might consider the Liouville measure of and the geodesic flow on $(T^*M, g^*)$.

From a different point of view, $g^*$ defines a smooth "bundle quadratic form" $Q^*$ on $T^*M$ : given $\alpha \in T^*_mM$, let $Q^*(\alpha) = \frac{1}{2} \, g^*(\alpha, \alpha)$. This is a smooth real function of $T^*M$, in fact a "Hamiltonian" function if we equip $T^*M$ with its canonical symplectic form. There is an associated Hamiltonian vector field $X_{Q^*}$ whose flow preserves the level sets of $Q^*$, that is, the sphere-subbundles of $T^*M$ of different $g^*$-radii. Furthermore, this flow happens to be the same as the above "geodesic flow" on $(T^*M, g^*)$.

Since this flow preserves both the canonical volume form on $(T^*M, \omega_0)$ and the Liouville measure on $(T^*M, g^*)$, both volume forms have to be proportional to one another by a (nonvanishing) function $f$ which is constant along the flow. A priori, there is no reason for $f$ to equal 1 : after all, the canonical volume form knows nothing about the metric $g$ on $M$. However, in order to restrict the canonical volume form to a volume form on the level sets of $Q^*$, one needs to take the interior product with (for instance) a normal vector field, which necessitates a Riemannian metric on $T^*M$. I don't know if it is true, but may be there is a way to get the same measure on the level sets from both approaches.

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If you take a look at the Wikipedia article on Liouville's theorem, you can read the following :

Although the equation is usually referred to as the "Liouville equation", Josiah Willard Gibbs was the first to recognize the importance of this equation as the fundamental equation of statistical mechanics. It is referred to as the Liouville equation because its derivation for non-canonical systems utilises an identity first derived by Liouville in 1838.

This suggests that the names "Liouville measure" might be only honorific too, since proofs of "the geodesic flow preserves the measure $\nu$" and "the Hamiltonian flow preserves the measure $\mu$" exist which use an identity of Liouville.