Canonical variables and geodesic fields for the calculus of variations of multiple integrals in parametric form

• DOI: 10.1007/bf01114915
• Publication Date: 2005-02-05T16:00:16Z
• Authors: D. H. Martin
• Keywords: 510.mathematics; Article

As is well known there remain many unsolved problems in the CaIculus of Variations of multiple integrals; in particular, the theory for integrals in parametric form has not received much attention. This paper presents a canonical formalism with a new Hamiltonian function developed especially for application to the theory of geodesic fields of CARATHI~ODORY [2], or more precisely, to the form assumed by this theory for parametric integrals. As is pointed out by RUND [5], the canonical formalism given by CARATn~ODORY is not applicable to this case. Part one of the paper contains some brief immediate consequences of the parameter-invariance of the fundamental integral, and the definition and properties of the canonical variables and Hamiltonian function; these latter quantities being introduced by means of a certain contact transformation. In part two, some special features of geodesic fields of parametric integrals are pointed out, and an explicit characterization of geodesic fields by means of a generalized Hamilton-Jacobi equation is established. Finally a theorem concerning the integrability conditions of the field is proved, which restores an initial lack of determinism of the slope functions of a field.