2015-77.md

course	course_year	question_number	tags	title	year
Variational Principles	IB	77	IB 2015 Variational Principles	Paper 1, Section I, A	2015

Consider a frictionless bead on a stationary wire. The bead moves under the action of gravity acting in the negative $y$-direction and the wire traces out a path $y(x)$, connecting points $(x, y)=(0,0)$ and $(1,0)$. Using a first integral of the Euler-Lagrange equations, find the choice of $y(x)$ which gives the shortest travel time, starting from rest. You may give your solution for $y$ and $x$ separately, in parametric form.