Isoperimetric problem

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