View source for Lindelöf construction

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

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/l058950a.gif" />

Figure: l058950a

Lindelöf's construction remains suitable for any variational problem of the simplest type on the $(x,y)$-plane for which the general integral of the [[Euler equation|Euler equation]] can be represented in the form

$$\frac{y}{c_1}=f\left(\frac{x-c_2}{c_1}\right).$$

The tangents to the extremals at conjugate points $A$ and $A'$ intersect at some point $T$ on the $x$-axis, and the value of the variable integral along the arc $AA'$ is equal to its value on the polygonal line $ATA'$ (see [[#References|[2]]]). An example is the [[Catenoid|catenoid]] with generating curve

$$\frac{y}{c_1}=\cosh\frac{x-c_2}{c_1}.$$

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Lindelöf,   "Leçons de calcul des variations" , Paris  (1861)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  O. Bolza,   ''Bull. Math. Soc.'' , '''18''' :  3  (1911)  pp. 107–110</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Carathéodory,   "Variationsrechnung und partielle Differentialgleichungen erster Ordnung" , Teubner  (1956)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  A.E. Bryson,   Y.-C. Ho,   "Applied optimal control" , Blaisdell  (1969)</TD></TR></table>

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