Math 6420 • PDE

The so called ``Brachistochrone problem'' admits such
formulation. ``A particle is sliding along a trough (in a vertical plane from the point
\((0,0)\) to some given point \((x_1,y_1)\) \((x_1>0,\ y_1<0)\) under the effect of gravity
without friction (having zero initial velocity). Find the form \(y=Y(x)\) of the trough which
minimizes the time of descent of this particle.''
\begin{enumerate}
\item[a).] Find the functional that ought to be minimized.
\item[b).] Using this functional, write the boundary value problem that ought to be solved
for finding the solution of the ``Brachistochrone problem''. You do not need to solve this
boundary value problem.
\end{enumerate}