Second Order Lagrangian Twist Systems: Simple Closed Characteristics

We consider a special class of Lagrangians that play a fundamental role in the theory of second order Lagrangian systems: Twist systems. This subclass of Lagrangian systems is deened via a convenient monotonicity property that such systems share. This monotonicity property (Twist property) allows a nite dimensional reduction of the variational principle for nding closed characteristics in xed energy levels. This reduction has some similarities with the method of broken geodesics for the geodesic variational problem on Riemannian manifolds. On the other hand, the monotonicity property c a n be related to the existence of local Twist maps in the associated Hamiltonian ow. The nite dimensional reduction gives rise to a second order monotone recurrence relation. We study these recurrence relations to nd simple closed characteristics for the Lagrangian system. More complicated closed characteristics will be dealt with in future work. Furthermore, we give conditions on the Lagrangian that guarantee the Twist property. 1. Introduction Various mathematical models for problems in nonlinear elasticity, nonlinear optics , solid mechanics, etc. are derived from second order Lagrangian principles, i.e., the diierential equations are obtained as the Euler-Lagrange equations of a Lagrangian L that depends on a state variable u, and its rst and second order derivatives. The Euler-Lagrange diierential equations are fourth order and are of conservative nature. In scalar models the Lagrangian action is deened by Ju] =