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One of the fundamental problems in the classical calculus of variations. The isoperimetric problem consists in minimizing a functional
 
One of the fundamental problems in the classical calculus of variations. The isoperimetric problem consists in minimizing a functional
  
<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/i/i052/i052880/i0528801.png" /></td> </tr></table>
+
$$
 +
J _ {0} ( y)  = \
 +
\int\limits _ { x _ {1} } ^ { {x _ 2 } }
 +
f _ {0} ( x, y, y  ^  \prime  )  dx
 +
$$
  
 
under constraints of the form
 
under constraints of 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/i/i052/i052880/i0528802.png" /></td> </tr></table>
+
$$
 +
J _ {i} ( y)  = \
 +
\int\limits _ { x _ {1} } ^ { {x _ 2 } }
 +
f _ {i} ( x, y, y  ^  \prime  ) \
 +
dx  = c _ {i} ;
 +
$$
  
<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/i/i052/i052880/i0528803.png" /></td> </tr></table>
+
$$
 +
f _ {i} : \mathbf R \times \mathbf R  ^ {n} \times \mathbf R  ^ {n}  \rightarrow  \mathbf R ,\  i = 1 \dots m,
 +
$$
  
 
and certain boundary conditions.
 
and certain boundary conditions.
  
The isoperimetric problem reduces to the [[Lagrange problem|Lagrange problem]] when new variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052880/i0528804.png" /> are introduced satisfying the differential equations
+
The isoperimetric problem reduces to the [[Lagrange problem|Lagrange problem]] when new variables $  z _ {i} $
 +
are introduced satisfying the differential equations
  
<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/i/i052/i052880/i0528805.png" /></td> </tr></table>
+
$$
 +
\dot{z} _ {i}  = f _ {i} ( x, y, y  ^  \prime  ),\ \
 +
i = 1 \dots m,
 +
$$
  
 
with boundary conditions
 
with boundary conditions
  
<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/i/i052/i052880/i0528806.png" /></td> </tr></table>
+
$$
 +
z _ {i} ( x _ {1} )  = 0,\ \
 +
z _ {i} ( x _ {2} )  = c _ {i} ,\ \
 +
i = 1 \dots m.
 +
$$
  
 
Necessary conditions for optimality in the isoperimetric problem have the same form as do the simplest problems in the calculus of variations related to the [[Lagrange function|Lagrange function]]
 
Necessary conditions for optimality in the isoperimetric problem have the same form as do the simplest problems in the calculus of variations related to the [[Lagrange function|Lagrange function]]
  
<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/i/i052/i052880/i0528807.png" /></td> </tr></table>
+
$$
 +
L ( x, y, y  ^  \prime  , \lambda _ {0} \dots \lambda _ {m} )  = \
 +
\sum _ {i = 0 } ^ { m }
 +
\lambda _ {i} f _ {i} ( x, y, y  ^  \prime  ).
 +
$$
  
 
The name  "isoperimetric problem"  goes back to the following classical question: Among all the curves with given perimeter in the plane, find the one that bounds the largest area.
 
The name  "isoperimetric problem"  goes back to the following classical question: Among all the curves with given perimeter in the plane, find the one that bounds the largest area.
Line 27: Line 62:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.A. Bliss,  "Lectures on the calculus of variations" , Chicago Univ. Press  (1947)  {{MR|0017881}} {{ZBL|0036.34401}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.Ya. Tslaf,  "Calculus of variations and integral equations" , Moscow  (1970)  (In Russian)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.A. Lavrent'ev,  L.A. Lyusternik,  "A course in variational calculus" , Moscow-Leningrad  (1950)  (In Russian)  {{MR|}} {{ZBL|}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.A. Bliss,  "Lectures on the calculus of variations" , Chicago Univ. Press  (1947)  {{MR|0017881}} {{ZBL|0036.34401}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.Ya. Tslaf,  "Calculus of variations and integral equations" , Moscow  (1970)  (In Russian)  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.A. Lavrent'ev,  L.A. Lyusternik,  "A course in variational calculus" , Moscow-Leningrad  (1950)  (In Russian)  {{MR|}} {{ZBL|}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
As stated above, the original isoperimetric problem is the problem of finding the geometric figure with maximal area and given perimeter. I.e., the problem is to find functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052880/i0528808.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052880/i0528809.png" /> such that
+
As stated above, the original isoperimetric problem is the problem of finding the geometric figure with maximal area and given perimeter. I.e., the problem is to find functions $  y _ {1} ( x) $,  
 +
$  y _ {2} ( x) $
 +
such that
  
<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/i/i052/i052880/i05288010.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { x _ {1} } ^ { {x _ 2 } } y _ {1} y _ {2}  ^  \prime  d x
 +
$$
  
 
is minimized, subject to
 
is minimized, subject to
  
<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/i/i052/i052880/i05288011.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { x _ {1} } ^ { {x _ 2 } }
 +
\sqrt {( y _ {1}  ^  \prime  )  ^ {2} + ( y _ {2}  ^  \prime  )  ^ {2} } \
 +
d x  = l ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052880/i05288012.png" /> is a given constant.
+
where $  l $
 +
is a given constant.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.E. [L.E. El'sgol'ts] Elsgolc,  "Calculus of variations" , Pergamon  (1961)  (Translated from Russian)  {{MR|0344552}} {{MR|0279361}} {{MR|0209534}} {{MR|1532560}} {{MR|0133032}} {{MR|0098996}} {{MR|0051448}} {{ZBL|0101.32001}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.E. [L.E. El'sgol'ts] Elsgolc,  "Calculus of variations" , Pergamon  (1961)  (Translated from Russian)  {{MR|0344552}} {{MR|0279361}} {{MR|0209534}} {{MR|1532560}} {{MR|0133032}} {{MR|0098996}} {{MR|0051448}} {{ZBL|0101.32001}} </TD></TR></table>

Revision as of 22:13, 5 June 2020


One of the fundamental problems in the classical calculus of variations. The isoperimetric problem consists in minimizing a functional

$$ J _ {0} ( y) = \ \int\limits _ { x _ {1} } ^ { {x _ 2 } } f _ {0} ( x, y, y ^ \prime ) dx $$

under constraints of the form

$$ J _ {i} ( y) = \ \int\limits _ { x _ {1} } ^ { {x _ 2 } } f _ {i} ( x, y, y ^ \prime ) \ dx = c _ {i} ; $$

$$ f _ {i} : \mathbf R \times \mathbf R ^ {n} \times \mathbf R ^ {n} \rightarrow \mathbf R ,\ i = 1 \dots m, $$

and certain boundary conditions.

The isoperimetric problem reduces to the Lagrange problem when new variables $ z _ {i} $ are introduced satisfying the differential equations

$$ \dot{z} _ {i} = f _ {i} ( x, y, y ^ \prime ),\ \ i = 1 \dots m, $$

with boundary conditions

$$ z _ {i} ( x _ {1} ) = 0,\ \ z _ {i} ( x _ {2} ) = c _ {i} ,\ \ i = 1 \dots m. $$

Necessary conditions for optimality in the isoperimetric problem have the same form as do the simplest problems in the calculus of variations related to the Lagrange function

$$ L ( x, y, y ^ \prime , \lambda _ {0} \dots \lambda _ {m} ) = \ \sum _ {i = 0 } ^ { m } \lambda _ {i} f _ {i} ( x, y, y ^ \prime ). $$

The name "isoperimetric problem" goes back to the following classical question: Among all the curves with given perimeter in the plane, find the one that bounds the largest area.

References

[1] G.A. Bliss, "Lectures on the calculus of variations" , Chicago Univ. Press (1947) MR0017881 Zbl 0036.34401
[2] L.Ya. Tslaf, "Calculus of variations and integral equations" , Moscow (1970) (In Russian)
[3] M.A. Lavrent'ev, L.A. Lyusternik, "A course in variational calculus" , Moscow-Leningrad (1950) (In Russian)

Comments

As stated above, the original isoperimetric problem is the problem of finding the geometric figure with maximal area and given perimeter. I.e., the problem is to find functions $ y _ {1} ( x) $, $ y _ {2} ( x) $ such that

$$ \int\limits _ { x _ {1} } ^ { {x _ 2 } } y _ {1} y _ {2} ^ \prime d x $$

is minimized, subject to

$$ \int\limits _ { x _ {1} } ^ { {x _ 2 } } \sqrt {( y _ {1} ^ \prime ) ^ {2} + ( y _ {2} ^ \prime ) ^ {2} } \ d x = l , $$

where $ l $ is a given constant.

References

[a1] L.E. [L.E. El'sgol'ts] Elsgolc, "Calculus of variations" , Pergamon (1961) (Translated from Russian) MR0344552 MR0279361 MR0209534 MR1532560 MR0133032 MR0098996 MR0051448 Zbl 0101.32001
How to Cite This Entry:
Isoperimetric problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Isoperimetric_problem&oldid=47444
This article was adapted from an original article by I.B. Vapnyarskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article