  In the context of mechanical Lagrangian dynamics, we prove a new Lyapunov
instability criterion for a non strict local minimum equilibrium point of a
smooth potential where the sufficient condition for instability is the
existence of a smooth solution of a certain linear PDE derived from the
mechanical Lagrangian governing the dynamics. In the presence of a
magnetostatic field, we also give an additional sufficient condition for the
motion of a charged particle to be Lyapunov unstable.
