The multiplier rule for ordinary differential equations

One of the unsolved problems of% the calculus of variations is to find a proof of the multiplier rule for the case of partial differential equations as side conditions.The essential difficulty lies in the fact that the theory of partial differential equations is not sufficiently developed to allow the use of the same proecedure which worked in the case of ordinary differential equations.Hence, it would be desirable to have a proof of the multiplier rule which made no appeal to the theory of differential equations, even in the case of ordinary differential equations.A proof of this type is the object of this paper.Let us consider the problem of making stationary the functional J[y] defined by F(x, yi ■ ■ ■ yn, yl ■ ■ ■ yñ )dx sounder boundary conditions which need not be specified here and under side conditionsThe usual proof of the multiplier rule starts from this problem and shows that the problem of making stationary the functional K[y, X],(3) K[y,\] = Ç* {F + \,Gi}dx, _