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 specially of the non-involutive type.Iam aware of Ivrii-type Fuchsian operators,already posing great difficulties. Should be grateful for inputs-Nagaraj Iyengar,community wiki

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.