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 partial differential equations 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 Poincaré–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 Poincaré–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.