A colleague posed a toy problem to me today that degenerates to finding the curve y(x) of shortest length than connects two points in the plane (WLOG: y(0) = 0, y(a) = b), such that y'(0) = 0.
This is of course very close to the classic introductory problem in the calculus of variations, except for the constraint on the slope at the left boundary.
From what I recall, a constraint such as this can be aggregated into the variational expression as:
$$\min_{y}: J = \int_0^a \sqrt{1 + y'^2} dx + \lambda y'(0) $$
What I don't recall is whether this can be addressed via the solving an associated Euler-Lagrange equation, or whether a similar equation describing a first-order condition needs to be derived from $\delta J$. A quick literature review & shuffling through my grad school notes didn't provide much insight for a constraint of this form (non-integral & at a single point). Any insight on this?
There is no shortest such curve. Start by tracing out a circle of radius $\epsilon$ centered at $(0,\epsilon)$ until you hit the tangent line to that circle which passes through $(a,b)$, and follow that line. As $\epsilon\to0$, you get a family of $y$'s all having $y'(0)=0$, whose arc lengths converge to that of the straight line.