Dynamical systems. Roughly speaking, a dynamical system $\dot{x} = a(x)$ is stable if and only if the 1st order linear partial differential equation $\mathcal{L}_a v + \ell = 0$ has a positive solution $v$. Here $v$ is called a Lyapunov function for the system, $\mathcal{L}$ is the Lie derivative, and $\ell > 0$ has to be chosen so that the equation has a solution but is otherwise arbitrary.