Linear partial differential operators (or, in the language of quantum mechanics, quantum observables) on, say, ${\bf R}^n$, are (in principle, at least) generated by the position operators $x_j$ and the momentum operators $\frac{1}{i} \frac{\partial}{\partial x_j}$, which are then related to each other by the basic commutation relations $$ \frac{1}{i} [x_j, \frac{1}{i} \frac{\partial}{\partial x_k}] = \delta_{jk}$$ where $[,]$ here is the commutator $[A,B] = AB-BA$.
Meanwhile, classical observables on the phase space $T^* {\bf R}^n$ are (again in principle) generated by the position functions $q_j$ and momentum functions $p_j$, which are related to each other by the basic commutation relations $$ \{ q_j, p_k \} = \delta_{jk}$$ where $\{,\}$ is now the Poisson bracket (I may have the sign conventions reversed here).
One of the great insights of quantum mechanics (or, on the mathematical side, semi-classical analysis) is the correspondence principle that asserts, roughly speaking, that the behaviour of quantum observables converges in the high-frequency limit (or, after rescaling, the semi-classical limit) to the analogous behaviour of classical observables. The correspondence is easier to see on the observable side than on the physical space side, for instance by connecting the von Neumann algebra of bounded quantum observables with smooth symbol with the Poisson algebra of smooth classical observables. The former is connected to linear PDE and the latter to symplectic (or Hamiltonian) geometry.
Another way to see the connection is to investigate what happens when one applies a linear partial differential (or pseudodifferential) operator to a high-frequency function (or "quantum state"), when viewing that function through its Wigner transform, which can be viewed as approximately describing the quantum state by a classical one. A standard calculation shows (under Weyl quantisation) that the top order contribution of the operator on this transform is given by its symbol, and the next order term is basically given by the Hamiltonian vector field associated to that symbol. (This is discussed for instance in Folland's "Harmonic analysis on phase space".) This suggests that the dynamics of linear PDE at high frequencies are going to be driven by the associated Hamiltonian dynamics of the symbol of that PDE.