  The classical problem of the brachistochrone asks for the curve down which a
body sliding from rest and accelerated by gravity will slip (without friction)
from one point to another in least time. In undergraduate courses on classical
mechanics, the solution of this problem is the primary example of the power of
the variational calculus. Here we address the generalized brachistochrone
problem that asks for the fastest sliding curve between a point and a given
curve or between two given curves. The generalized problem can be solved by
considering variations with varying endpoints. We will contrast the formal
solution with a much simpler solution based on symmetry and kinematic
reasoning. Our exposition should encourage teachers to include variational
problems with free boundary conditions in their courses and students to try
simple, intuitive solutions first.

Classical Physics