Using the mechanical principle that the centre of mass
places itself as low as possible, determine the equation of the curve formed by a flexible homogeneous wire or a thin chain with length
when supported at its ends in the points
and
.
We have an isoperimetric problem
to minimise![]() |
(1) |
![]() |
(2) |
![]() ![]() |
(3) |
The Euler-Lagrange differential equation, the necessary condition for (3) to give an extremal
, reduces to the Beltrami identity
where
which may become clearer by notating
we choose the new constant of integration
We can write two equivalent results
i.e.
Adding these allows to eliminate the square roots and to obtain
or
![]() |
(4) |
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