On a smooth manifold of dimension $n$, the application value of the canonical $1$-form, the Liouville form on $T^*(X)$, to the Hamiltonian mechanics is well known; $T^*(X)$ is a degree $1$-Jet bundle. My question is *Do canonical forms similar to the Liouville form exist on higher degree Jet bundles?*
I ask this because, beyond the invariant sub-principal symbol of a pseudodifferential operator, nothing much seems to be known to handle multiple characteristic problems, especially of the non-involutive
type. I am aware of Ivrii-type Fuchsian operators, already posing great difficulties.