If one views $\Bbb R ^{2n}$ as the cotangent bundle of $\Bbb R ^n$, with coordinates $(q_1, \dots, q_n, p_1, \dots, p_n)$, then in order to do classical Hamiltonian mechanics on it one considers classical observables which are polynomial functions in $q$ and $p$.
If one considers now an abstract smooth manifold (endowed with whatever supplementary structure makes your life easier, such as a Riemannian or a symplectic one), can one come up with an analogous concept to the one of "polynomial function"?