Symplectic structures related with higher order variational problems

Abstract

In this paper we derive the symplectic framework for field theories defined by higher-order Lagrangians. The construction is based on the symplectic reduction of suitable spaces of iterated jets. The possibility of reducing a higher-order system of PDEs to a constrained first-order one, the symplectic structures naturally arising in the dynamics of a first-order Lagrangian theory, and the importance of the Poincar\'e-Cartan form for variational problems, are all well-established facts. However, their adequate combination corresponding to higher-order theories is missing in the literature. Here we obtain a consistent and truly finite-dimensional canonical formalism, as well as a higher-order version of the Poincar\'e-Cartan form. In our exposition, the rigorous global proofs of the main results are always accompanied by their local coordinate descriptions, indispensable to work out practical examples.