Lyapunov’s Direct Method for Non-strict Minima

Indianapolis, IN

According to Lyapunov's Direct Method, the strict local minimum of a (negative definite) Lyapunov function of a dynamical system is a (asymptotically) stable equilibrium. In a Hamiltonian system, the Lyapunov function can be chosen to be the Hamiltonian of the system. For a conservative mechanical system, in particular, this implies that a position of strict local minimum of the potential corresponds to a stable equilibrium state. In this article, we demonstrated that a position of local minimum, not necessarily strict, of the (negative definite) Lyapunov function also implies (asymptotic) stability under an intuitive subsidiary condition. Using this, we showed that for a conservative mechanical system the local minimum, not necessarily strict, of the potential corresponds to a stable equilibrium. We applied this to the case of a general conservative mechanical system with one degree of freedom. For this system, we also rigorously derived the type of motion in the vicinity of a stable equilibrium.