| 释义 |
Sphere with TunnelFind the tunnel between two points  and  on a gravitating Sphere which gives the shortest transit time under theforce of gravity.  Assume the Sphere to be nonrotating, of Radius  , and withuniform density  . Then the standard form Euler-Lagrange Differential Equation in polarcoordinates is
 | (1) |
 ,  ,  , and  . Integrating once gives
 | (2) |
 rollinginside the Circle of Radius , so the tunnel is shaped like an arc of a Hypocycloid. The transit timefrom point to point is
 | (3) |
 | (4) |
 the universal gravitational constant.
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