On the Carathéodory Form in Higher-Order Variational Field Theory

Abstract

The Carathéodory 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 intrinsic geometric operations applied to the well-known Poincaré–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 Carathéodory equivalent of the Hilbert Lagrangian in general relativity is discussed.

The Fundamental Lepage Form in Two Independent Variables: A Generalization Using Order-Reducibility

“…We summarize basic facts about Lepage differential forms on finite-order jet prolongations of fibered manifolds and, in particular, we discuss distinguished examples of Lepage equivalents of first-and second-order Lagrangians; for more details see [6,8,20,[22][23][24].…”

“…The expression Λ λ (11) is the well-known Carathéodory form, associated to a nonvanishing, first-order Lagrangian λ (cf. [3]), whereas Λ λ (12) is its generalization for secondorder Lagrangians, recently studied in [8].…”

“…Lepage forms play a basic role in the calculus of variations of both simple-and multipleintegral problems over fibered manifolds and Grassmann fibrations. Among the well-known examples of Lepage forms, we note: the Cartan form of classical mechanics and its generalization in higher-order mechanics (Krupka [1]); in first-order field theory, the Poincaré-Cartan form (García [2]), the Carathéodory form (Carathéodory [3]), and the fundamental Lepage form (also known as the Krupka-Betounes form) (Krupka [4], Betounes [5]); in secondorder field theory, the generalized Poincaré-Cartan form (Krupka [6]), the generalized Carathéodory form (Crampin and Saunders [7], Urban and Volná [8]), the fundamental Lepage form for second-order, homogeneous Lagrangians (Saunders and Crampin [9]). See also Gotay [10], Goldschimdt and Sternberg [11], Rund [12], Dedecker [13], Horák and Kolář [14], Krupka [6], Krupka and Štěpánková [15], Saunders [16], and Sniatycki [17].…”