In Lectures on Partial Differential Equations, Arnold gives a theorem in the chapter Huygens' Principle in the Theory of Wave Propagation:
Theorem 1 (the theory of support functions). The manifold of 1-jets of functions from the sphere $S^{n-1}$ into $\mathbb R$ (with its natural contact structure) is contact diffeomorphic to the manifold of co-oriented contact elements in $\mathbb R^n$ (with its tautological contact structure).
By the manifold of 1-jets $J^1(B^n,\mathbb R)$ of a manifold $B^n$, he means the totality of $(x,p,y)$ where $x\in B^n,y\in\mathbb R$ and $(x,p)$ lies in the cotangent bundle of $B^n$. By the natural contact structure, he means Cartan distribution, i.e. approximately $dy=pdx$ if some local coordinate system is given. By tautological contact structure, he means the contact structure given by tautological 1-forms on the manifold of co-oriented contact elements, and by co-oriented, he means the contact elements are oriented, i.e., the manifold is $ST^*M$ rather than $PT^*M$ in the wiki page (in place of $\exists\lambda\ne0$, it's $\exists\lambda>0$). It's a rather abstract term, so I haven't understood well yet.
In the proof, there's a diffeomorphism from 1-jet space to the cotangent bundle: $(q\in S^{n-1},p=df\rvert_q\in(\mathbb R^{n-1})^*,z=f(q))\mapsto(Q=q+pz,P=q)$. I doubt it's wrong since I cannot see any way to restore $(q,p,z)$ from $(P,Q)$. I think it should be something similar to Legendre transformation, but I cannot explicate it.
Any idea? Thanks!
