On the transversal distribution and the complete figure in the second order multiple integral problem in the calculus of variations (Q5902699)

This paper addresses the calculus of variations of m-fold integrals with n field variables, and Lagrangian L depending on derivatives of up to the second order of the field variables. While the usual assumption is made that \(L>0\) in its domain of definition, a novelty of the theory presented here is that the condition is waived that the determinant of the Hamiltonian complex is nonvanishing. The approach being followed corresponds to one recently developed for the first order case by \textit{H. Rund} [Quaest. Math. 5, 187-218 (1982; Zbl 0491.49019)]. The transversal distribution plays a central role, together with its integral surfaces, the transversal subspaces. Independent integrals and a quasi-geodesic field are introduced, after which a geodesic field theory is constructed, where the author has dispensed with a rather restrictive symmetry condition on the functions \(S^ j\), \(j=1,...,n\), which give rise to such a field. This condition had been imposed in earlier work on the canonical theory for this problem by \textit{H. Rund} [Tydsk Natuurwetenskap 10, 228- 240 (1970)], and in a construction of the complete figure by the reviewer [Aequationes Math. 14, 363-386 (1976; Zbl 0348.49016)], while the reviewer and the author [Tensor, New Ser. 39, 240-248 (1982; Zbl 0514.49014)] subsequently already had indicated that a geodesic field theory can be constructed without assuming this condition to hold. Its influence on the existence conditions for the transversal subspaces is then discussed, and it is shown how all these results can be tied together in a complete figure in the sense of Carathéodory. When the symmetry condition is met, the theory contains those of Rund and of the reviewer as special cases. In the concluding sections of the paper, this symmetry condition is imposed once more, in the derivation and analysis of conditions which ensure that a quasi-geodesic field is rendered a geodesic one. Whereas most of the calculations and results are rather involved, the author has succeeded in presenting an eminently readable paper, avoiding unnecessary manipulative detail, reference for some of which being made to his Ph. D. thesis [Transversality and the complete figure in the second order multiple integral problem in the calculus of variations, University of South Africa, Pretoria (1982)].