[Related but different: https://math.stackexchange.com/questions/2989908/which-shape-does-an-elastic-rod-take-as-its-ends-are-getting-closer - different functional]
I'm looking for a shape that elastic beam makes when its ends are brought into contact. The energy functional in this case is
$$ \mathcal{E} = \int \left(\frac{1}{\rho}\right)^2 ds $$
where $\rho$ is radius of curvature. (We're looking for a curve that minimizes the energy).
How would one go about finding such curve? Is this a well known shape?
Rough picture of what it should look like for reference (if you have a piece of paper you can just bend it to see that the solution is not a circle like some would expect):
If we parametrize the tangent vector with angle such that
$$ \mathbf{T} = (\cos \theta , \sin \theta) $$
Then we need to minimize
$$ \int {\dot\theta}^2 ds $$
This becomes a variational problem and we impose boundary conditions by demanding that $x$ and $y$ coordinates return to zero (maybe this can be done more elegantly in different coordinates?) via Lagrange mulipliers. Resulting lagrangian is
$$ \mathcal{L} = \dot\theta^2 + \mu \cos \theta + \mu \sin \theta$$
plug into Euler-Lagrange equation to get ODE (with no explicit solution)
$$ 2 \ddot \theta = \mu \cos \theta - \lambda \sin \theta$$