On the Legendre map in higher-order field theories

Abstract

The authors show how the construction of a Cartan form in higher-order field theories defines a Legendre map, and how the regularity of this map may be described in terms of a sequence of maps between intermediate phase spaces. They also show how semi-holonomic jets are important in this context: they allow the definition of regularity to be applied without ambiguity to the Lagrangian itself, and they permit the specification of a unique Cartan form, Legendre map and covariant phase space for second-order Lagrangians.