Say, we have a Hamiltonian $H(x,p)$. We find a function $G(x,p)$ such that the function $H(x,p)+iG(x,p)$ has a complex derivative. $G$ is then the harmonic conjugate of $H$.
Since the change in $H(x,p)$ in time-evolution is 0 (assuming energy is conserved), the changes in $G(x,p)$ should be extremised during time evolution, because the time evolution is orthogonal to the gradient of $H(x,p)$, and hence is along the gradient of $G(x,p)$. This sounds a lot like the Lagrangian (as the action, just like $\delta G(x,p)$, is extremised during time evolution). Does $G$ have anything to do with the lagrangian? What is $G$? Why are we never taught about it?
