### The dependence of focal points upon curvature for problems of the calculus of variations in space

Marion Ballantyne White
1912 Transactions of the American Mathematical Society
In the Calculus of Variations, if an arc Coi which joins a space curve L and a fixed point 1 minimizes the integral (1) J = J f(x, y,y', z, z')dx with respect to other curves joining L with 1, there will in general be a focal point 2 lying beyond 1 on the curve C of which C0i is a part, at which the minimizing property ceases. For the space problem it is well known that the minimizing arc Coi must an extremal and must be cut by L transversally at their point of intersection 0 .f If these
more » ... .f If these conditions are satisfied, C0i can be imbedded in a two-parameter family of extremals to each of which L is transversal and which will have an enveloping surface. If the enveloping surface has no singular point at its contact point with C, a further necessary condition for Coi to be a minimizing arc is that this contact point, which is the focal point mentioned above, does not lie between 0 and 1. These results can be derived with the help of geometrical considerations, but the geometrical methods fail in case the enveloping surface has a singular point at its contact with C0i. It is the purpose of the present paper to derive the properties of the focal point by means of the second variation, and to show that they persist even when the enveloping surface may have a singularity. Further, the dependence of the position of the focal point upon the curvature of L will be discussed, and a result derived which is analogous to that which has already been found for the corresponding problem in the plane.J It is found that if a direction p, q, r through the intersection point 0 is properly chosen, and if t is the length of the segment in this direction which projects orthogonally into the radius of curvature of L at 0, then the distance from 0