Jacobi’s condition for problems of the calculus of variations in parametric form

There are two well-known methods of deducing Jacobi's necessary condition in the calculus of variations. One is geometric in character, depending upon a property of an envelope of a one-parameter family of extremals through a fixed point, the cases when the envelope has a singular point being usually excluded.f The second proof involves complicated manipulations of the second variation. For the problem in parametric form in the plane the reduction of the second variation was devised by Weierstrass and is a remarkable piece of analysis. J It is, however, very artificial and not easily extensible to problems in more than two dimensions. For problems in parametric form in higher spaces a discussion of the second variation has been made by von Escherich§ by methods in part quite unsymmetrical. The lack of symmetry is due to the division of an arc Vi = yi{t) «iSisiiii-1,2, ••-,»»)