isoperimetric problem

The simplest of the isoperimetric problems is the following:

One must set an arc with a given length $l$ from a given point $P$ of the plane to another given point $Q$ such that the arc together with the line segment $P\mathit{}Q$ encloses the greatest area possible.

This task is solved in the entry example of calculus of variations.

More generally, isoperimetric problem may determining such an arc $c$ between the given points $P$ and $Q$ that it gives for the integral

${\displaystyle {\int }_P^Q}f(x,y,y^{\prime })𝑑s$(1)

an extremum and that gives for another integral

${\displaystyle {\int }_P^Q}g(x,y,y^{\prime })𝑑s$(2)

a given value $l$, as both integrals are taken along $c$.  Here, $f$ and $g$ are given functions.

The constraint (2) can be omitted by using the function $f-\lambda g$ instead of $f$ in (1) similarly as in the mentionned example.