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.



Canonical forms on higher degree Jet bundles similar to the Liouville form
The $k$-jet bundle $J^k$ of $k$-jets of real valued functions on a manifold $M$ has an obvious map $J^k\to J^1$, if $k\ge 1$, smooth and diffeomorphism invariant, taking the $k$-jet of a function to its $1$-jet. The $1$-jet bundle has an obvious splitting $J^1=J^0 \oplus T^\* X$, mapping each $1$-jet to its $0$-jet and exterior derivative of any function, as calculated from its $1$-jet. Pull back the Liouville form $\lambda$ from $T^\*X$ to $J^1$, and to $J^k$, to obtain a $1$-form $\lambda$ on $J^k$ so that, for any $C^{k+1}$ function $f$, its $k$-jet $j^k f$, as a section of $J^k$, satisfies $(j^k f)^\*\lambda=df$. Is that what you are looking for?


