The invariant integral and the inverse problem in the calculus of variations

Abstract

must be taken in order that the value of the integral may be an extremum. The differential equation which defines the desired curve is of the second order and its general solution is a two-parameter family of curves known as the extremals of the problem. By the successive substitution of the coordinates of the end points, P1 and P2, in the equation of the extremals we obtain two equations for the determination of the values of the parameters in terms of these coordinates. 2. The inverse problem is, as its name suggests, that of determining the form of the integrand function which has as its extremals a given twoparameter family of curves. 3. Besides the simplest problem as we have stated it there is the problem of determining the extremals of an integral,