  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.
