Dido was given a 50 meter hide rope to enclose maximum land area adjoining a sea coast.
$$ \int y ~dx - \lambda \int \sqrt{1+y^{'2}} ~dx;~$$
$$L=y- \lambda \sqrt{1+y^{'2}} $$
From Euler-Lagrange equation, $x$ does not explicitly occur.
$$ y-\lambda \sqrt{1+y^{'2}} - y'\{ 0-\lambda \frac{y'}{ \sqrt{1+y^{'2}}}\}=h $$
where $h$ is an arbitrary integration constant.
Simplifying, since $\sec \phi= \sqrt{1+y^{'2}},~~$ we arrive at the ode
$$\cos \phi=\dfrac{y-h}{\lambda}$$
There are two constants $(\lambda,h)$. The Lagrange multiplier $\lambda$ of this isoperimetric problem is recognized geometrically as the semi-circle radius and constant $h$ as a rectangle height onto which the semi-circle is shifted up along the y-axis. The rectangle has sides $(2\lambda,h)$.
Several pairs $(\lambda,h) $ can be chosen with constant perimeter length
$$ \pi \lambda +2 h =50 $$
out of which three options are
$$ \{\lambda, h\}~=(15.955,10,0), (0, 9.292,25)$$
from which Dido cleverly chooses first obvious intuitive option $h=0$.
The second and last options are discarded as the object function area itself is compromised.
In this geometry situation a choice of $h=0$ is easy to make.
Should $h$ be always taken zero in such odes ?
An argument that the Lagrangian constant $h$ in the Euler Lagrange equation in all such situations should be set be zero seems unacceptable.
In another isoperimetric problem seeking solution of maximum area enclosed for give surface area involving such an arbitrary but important constant $c$,
$$ \int \pi r^2 dz - \lambda_1\int 2 \pi r \sqrt{1+r^{'2}} dz$$ which can be also reformed to Lagrangian
$$ r^2-\lambda r \sqrt{1+r^{'2}}$$
leading to ode using EL equation $$ \cos \phi =\frac{r^2-c^2}{r \lambda} $$
We recognize mean curvature $ H =\dfrac{1}{\lambda} $ for DeLaunay loop surfaces. Constant $c$ is distance between symmetry axis and points of reentrant tangency, the red and yellow lines.
If we set $c=0$ then all these loops reduce to spheres of radius $\lambda.$
Thanks for all comments applicable in a wider context for choosing/interpreting this arbitrary constant.
