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
![\begin{displaymath}
r_{\phi\phi}(r^3-ra^2)+{r_\phi}^2(2a^2-r^2)+a^2r^2 = 0,
\end{displaymath}](s2_839.gif) |
(1) |
along with the boundary conditions
,
,
, and
.
Integrating once gives
![\begin{displaymath}
{r_\phi}^2={a^2r^2\over {r_0}^2}{r^2-{r_0}^2\over a^2-r^2}.
\end{displaymath}](s2_844.gif) |
(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
![\begin{displaymath}
T=\pi\sqrt{a^2-{r_0}^2\over ag},
\end{displaymath}](s2_846.gif) |
(3) |
where
![\begin{displaymath}
g={GM\over a^2}={\textstyle{4\over 3}}\pi\rho G a
\end{displaymath}](s2_847.gif) |
(4) |
is the surface gravity with
the universal gravitational constant.
© 1996-9 Eric W. Weisstein
1999-05-26