On the Carath\'eodory form in higher-order variational field theory

The Carath\'eodory form of the calculus of variations belongs to the class of Lepage equivalents of first-order Lagrangians in field theory. Here, this equivalent is generalized for second- and higher-order Lagrangians by means of intrisic geometric operations applied to the well-known Poincar\'e--Cartan form and principal component of Lepage forms, respectively. For second-order theory, our definition coincides with the previous result obtained by Crampin and Saunders in a different way. The Carath\'eodory equivalent of the Hilbert Lagrangian in general relativity is discussed.