= First Hamiltonian structure of the KdV equation
{title2=$J=-2\partial_x$}
One normalization of the first <Hamiltonian> structure uses the constant differential <Poisson operator> $J=-2\partial_x$. For <functionals> with <variational derivatives> $f,g$, its <Poisson bracket> is $\int fJg\,dx=-2\int fg'\,dx$ under periodic or vanishing-boundary conventions. Thus the field bracket is $\{u(x),u(y)\}=-2\partial_x\delta(x-y)$. The factor and sign change with normalization; the constant first-order operator is the invariant structural feature.
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