 Lyapunov functional

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.