On the Globalization of Lagrangian and Hamiltonian Formalisms in Higher Order Mechanics

The Lagrangian and Hamiltonian formalisms for higher order mechanics 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 in a generalized framework, which extends to higher orders a framework due to Tulczyjew in the case of first order theories, and which allows to deal also with cases in which the Lagrangian is not "regular". The global Poincaré-Cartan form is also discussed.