Lyapunov functional
= Lyapunov functional
{c}
A Lyapunov functional assigns a scalar to the entire state of an evolution equation and is monotone along its trajectories. For a <gradient flow> $A_T=\delta V/\delta\overline A$, the identity $dV/dT=2\int|A_T|^2$ follows from the variational <chain rule>, with boundary terms removed by periodicity or decay. A spatial density is not itself a Lyapunov functional unless all remaining coordinates are integrated or averaged.