Let $M$ be a manifold and let $P$ be a partial differential operator acting on $C^{\infty}(M)$. Associated to $P$ there's the obvious linear PDE: $$P(f)=0$$

Naturally associated with $P$ we also have a hamiltonian system $\Phi= (T^*M, \omega,H=\sigma(P))$ where the symplectic form $\omega$ is the standard one and principal symbol $\sigma(P)$ of $P$ is taken as the hamiltonian.

Question: What does the dynamics of $\Phi$ tell us about the original differential equation?

Obviously since we are only considering the principal symbol we won't get a terrible amount of information. On the other hand we are not taking only the zero locus $\{ \sigma(P)=0\}$ (AKA the characteristic variety / set ) so one might hope that we can at least find some information in $\Phi$ which is not already present in the geometry of $\{\sigma(P)=0\}$.

Disclaimer: Let me apologize in advance for asking this slightly vague question

Let $M$ be a manifold and let $P$ be a partial differential operator acting on $C^{\infty}(M)$. Associated to $P$ there's the obvious linear PDE: $$P(f)=0$$

Naturally associated with $P$ we also have a hamiltonian system $$\Phi= (T^*M, \omega,H=\sigma(P))$$ where the symplectic form $\omega$ is the standard one and principal symbol $\sigma(P)$ of $P$ is taken as the hamiltonian.

Question: What does the dynamics of $\Phi$ tell us about the original differential equation?

Obviously since we are only considering the principal symbol we won't get a terrible amount of information. On the other hand we are not taking only the zero locus $\{ \sigma(P)=0\}$ (AKA the characteristic variety / set ) so one might hope that we can at least find some information in $\Phi$ which is not already present in the geometry of $\{\sigma(P)=0\}$.