Poisson operator ($\{F,G\}=\int \frac{\delta F}{\delta u}J\left(\frac{\delta G}{\delta u}\right)dx$)

For field functionals, a Poisson operator $J$ is an operator whose displayed pairing of variational derivatives defines a Poisson bracket. It must be skew under the integral pairing and satisfy the functional Jacobi identity. A constant skew differential operator, such as $J={\partial }_x$ with periodic or vanishing boundary conditions, gives the simplest examples. The Hamiltonian evolution is $u_t=J(\delta H/\delta u)$.