Sphere with Tunnel
Find the tunnel between two points A and B on a gravitating sphere which gives the shortest transit time under the force of gravity. Assume the sphere to be nonrotating, of radius a, and with uniform density rho. Then the standard form Euler-Lagrange differential equation in polar coordinates is
| r_(phiphi)(r^3-ra^2)+r_phi^2(2a^2-r^2)+a^2r^2=0, |
(1)
|
along with the boundary conditions r(phi=0)=r_0, r_phi(phi=0)=0, r(phi=phi_A)=a, and r(phi=phi_B)=a. Integrating once gives
But this is the equation of a hypocycloid generated by a circle of radius (a-r_0)/2 rolling inside the circle of radius a, so the tunnel is shaped like an arc of a hypocycloid. The transit time from point A to point B is
where
| [画像: g=(GM)/(a^2)=4/3pirhoGa ] |
(4)
|
is the surface gravity with G the universal gravitational constant.
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Cite this as:
Weisstein, Eric W. "Sphere with Tunnel." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SpherewithTunnel.html