This is exactly right except that the space in which Liouville’s theorem holds is called phase space. Phase space is the cotangent bundle over configuration space; i.e., if the configuration space is an n-dimensional manifold M, then for every point in M there is a copy of an n-dimensional vector space. These n-vectors* represent momenta, and both a configuration and a momentum are necessary to uniquely specify a state of a classical (Hamiltonian) system.
* More precisely, they are one-forms—linear functions of n-vectors; i.e. they eat n-vectors and spit out scalars. One-forms are also called covariant vectors, whence the other kind are called contravariant. They are dual to each other (for a given n), and thus (contravariant) vectors can equivalently be considered linear functions of one-forms instead.
I think it would be nice if the post were edited to reflect this distinction. It wouldn’t take much effort; just a sentence inserted at the point where it switches from talking about configuration space to talking about phase space, and appropriate tweaks to a few subsequent sentences. The Wikipedia article Configuration space links here, by the way.
This is exactly right except that the space in which Liouville’s theorem holds is called phase space. Phase space is the cotangent bundle over configuration space; i.e., if the configuration space is an n-dimensional manifold M, then for every point in M there is a copy of an n-dimensional vector space. These n-vectors* represent momenta, and both a configuration and a momentum are necessary to uniquely specify a state of a classical (Hamiltonian) system.
* More precisely, they are one-forms—linear functions of n-vectors; i.e. they eat n-vectors and spit out scalars. One-forms are also called covariant vectors, whence the other kind are called contravariant. They are dual to each other (for a given n), and thus (contravariant) vectors can equivalently be considered linear functions of one-forms instead.
I think it would be nice if the post were edited to reflect this distinction. It wouldn’t take much effort; just a sentence inserted at the point where it switches from talking about configuration space to talking about phase space, and appropriate tweaks to a few subsequent sentences. The Wikipedia article Configuration space links here, by the way.