Variable end points problems in the calculus of variations : Coupled points

Abstract

In the literature the study of conjugate or focal points, for problems in the calculus of variations, has been restricted to the case when one of the two endpoints is fixed. The main goal of this paper is to provide for the general case, that is, when both are varying, a new definition for c to be coupled with b.

The new notion reduces to the classical one of conjugate or focal point when x(a) is fixed. We will also recall the notion of regularity that takes into account the fact that there could be nonzero constant functions admissible for the accessory problem. Finally we relate this necessary condition to the existence of a solution of a certain Riccati matrix differential equation.

The authors wish to thank G.N.A.F.A. of C.N.R. (Italy) and N.S.E.R.C. (Canada) whose financial support made this research possible