## Sphere with Tunnel

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.