Symplectic structures related with higher order variational problems

TL;DR

This paper develops a finite-dimensional symplectic formalism for higher-order field theories, extending classical structures like the Poincaré-Cartan form to higher-order Lagrangians using symplectic reduction of jet spaces.

Contribution

It introduces a consistent, global canonical formalism for higher-order variational problems, filling a gap in the literature by combining symplectic reduction with higher-order Lagrangian structures.

Findings

Derived a symplectic framework for higher-order Lagrangian field theories

Established a higher-order Poincaré-Cartan form

Provided both global proofs and local coordinate descriptions

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.