We consider a class of periodic solutions of second order difference equations with symplectic structure. We obtain an explicit condition for their stability in terms of the 4-jet of the generating function. This result can be seen as a Lagrangian counterpart of the problem of Lyapunov stability of fixed points of area-preserving diffeomorphisms. An application is given to the model of a bouncing ball. Joint work with Rafael Ortega. Note:This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023–2027)