Variational derivative

For a functional $H[u]$, its variational derivative is defined byafter integrating derivatives of $v$ by parts and discarding the boundary terms.

For a complex field $\psi$, the canonical Hamiltonian evolution conventionis accompanied by its complex conjugate equation.

An evolution equation has Hamiltonian form when a Hamiltonian operator applied to the variational derivative of a functional gives its time derivative. The induced functional bracket must satisfy the Poisson properties, including Jacobi.