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A geometrical construction to find conjugate points in the problem of finding a minimal surface of revolution (see Fig.).
 
A geometrical construction to find conjugate points in the problem of finding a minimal surface of revolution (see Fig.).
  
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Figure: l058950a
 
Figure: l058950a
  
Lindelöf's construction remains suitable for any variational problem of the simplest type on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058950/l0589501.png" />-plane for which the general integral of the [[Euler equation|Euler equation]] can be represented in the form
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058950/l0589502.png" /></td> </tr></table>
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$$\frac{y}{c_1}=f\left(\frac{x-c_2}{c_1}\right).$$
  
The tangents to the extremals at conjugate points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058950/l0589503.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058950/l0589504.png" /> intersect at some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058950/l0589505.png" /> on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058950/l0589506.png" />-axis, and the value of the variable integral along the arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058950/l0589507.png" /> is equal to its value on the polygonal line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058950/l0589508.png" /> (see [[#References|[2]]]). An example is the [[Catenoid|catenoid]] with generating curve
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058950/l0589509.png" /></td> </tr></table>
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$$\frac{y}{c_1}=\cosh\frac{x-c_2}{c_1}.$$
  
 
====References====
 
====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></table>
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<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>
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<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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====Comments====
 
 
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.E. Bryson,  Y.-C. Ho,  "Applied optimal control" , Blaisdell  (1969)</TD></TR></table>
 

Latest revision as of 19:28, 26 March 2023

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 $(x,y)$-plane for which the general integral of the 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 [2]). An example is the catenoid with generating curve

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

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


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How to Cite This Entry:
Lindelöf construction. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lindel%C3%B6f_construction&oldid=16519
This article was adapted from an original article by V.V. Okhrimenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article