Lindelöf construction

A geometrical construction to find conjugate points in the problem of finding a minimal surface of revolution (see Fig.).

Figure: l058950a

Lindelöf's construction remains suitable for any variational problem of the simplest type on the -plane for which the general integral of the Euler equation can be represented in the form

The tangents to the extremals at conjugate points and intersect at some point on the -axis, and the value of the variable integral along the arc is equal to its value on the polygonal line (see [2]). An example is the catenoid with generating curve

References

[1]	E. Lindelöf, "Leçons de calcul des variations" , Paris (1861)
[2]	O. Bolza, Bull. Math. Soc. , 18 : 3 (1911) pp. 107–110
[3]	C. Carathéodory, "Variationsrechnung und partielle Differentialgleichungen erster Ordnung" , Teubner (1956)

References

[a1]	A.E. Bryson, Y.-C. Ho, "Applied optimal control" , Blaisdell (1969)