= Poisson operator
{c}
{title2=$\{F,G\}=\int\frac{\delta F}{\delta u}\,J\!\left(\frac{\delta G}{\delta u}\right)dx$}
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 $u_t=J(\delta H/\delta u)$.
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