Find the tunnel between two points 
and 
on a gravitating sphere which gives
the shortest transit time under the force of gravity. Assume the sphere
to be nonrotating, of radius 
, and with uniform density 
. Then the standard form Euler-Lagrange
differential equation in polar coordinates is
 |
(1)
|
along with the boundary conditions 
, 
, 
, and 
. Integrating once gives
 |
(2)
|
But this is the equation of a hypocycloid generated by a circle of radius 
rolling inside the circle
of radius 
, so the tunnel is shaped like an arc of a hypocycloid.
The transit time from point 
to point 
is
 |
(3)
|
where
 |
(4)
|
is the surface gravity with 
the universal gravitational constant.
