Second-order Lagrangians admitting a first-order Hamiltonian formalism

TL;DR

This paper explores second-order Lagrangians that can be reformulated into a first-order Hamiltonian framework, applying the approach to Einstein-Hilbert gravity and BF theory, thus providing a natural setting for general relativity.

Contribution

It introduces a new notion of regularity to develop a first-order Hamiltonian formalism for certain second-order Lagrangians and discusses their equivalence and applications to gravity theories.

Findings

A new Hamiltonian formalism for specific second-order Lagrangians is established.

The formalism is successfully applied to Einstein-Hilbert Lagrangian and BF theory.

The results support the naturalness of this variational setting for general relativity.

Abstract

Second-order Lagrangian densities admitting a first-order Hamiltonian formalism are studied; namely, i) for each second-order Lagrangian density on an arbitrary fibred manifold $p\colon E\to N$ the Poincar\'e-Cartan form of which is projectable onto $J^1E$, by using a new notion of regularity previously introduced, a first-order Hamiltonian formalism is developed for such a class of variational problems; ii) the existence of first-order equivalent Lagrangians are discussed from a local point of view as well as global; iii) this formalism is then applied to classical Einstein-Hilbert Lagrangian and a generalization of the BF theory. The results suggest that the class of problems studied is a natural variational setting for GR.