On the Global Structure of Lagrangian and Hamiltonian Formalisms in Higher Order Calculus of Variations

The Lagrangian and Hamiltonian formalisms for higher order calculus of variations on r-th order jet bundles are discussed in detail on the basis of the theory of pre-symplectic fibered manifolds. The concepts of Legendre mapping and regularity, and the relations existing between the Lagrangian and the Hamiltonian global equations are discussed. This framework extends to higher orders a framework already introduced by Tulczyjew in the case of first order theories; it allows to deal also with cases in which the Lagrangian is not "regular".