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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|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|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====
<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====
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====
<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>