Underground from Portland to New York (Part 2)
This post is part two of what will be a three part series on the
derivation of the equations of an underground tunnel that minimizes
travel time when only gravity is used for propulsion.
We left our subterranean tunnel problem with the following equation we
wished to minimize.
$$
\eqalign{
I &= \int_a^b\frac{\sqrt{(dr)^2+(rd\theta)^2}}{\sqrt{\frac{g}{R}(R^2-r^2)}}
}
$$
Our first issue is figuring out what to do with \(\sqrt{(dr)^2+(rd\theta)^2}\)
so that we can integrate it. We could choose to factor out either the
\(dr\) or the \(d\theta\) and could solve the problem either way. We'll
choose to factor out the \(dr\) because that leaves us an integrand that
is not a function of the path variable (in this case \(\theta\)) which
means that \(\pard{F}{\theta} = 0\) and \(\pard{F}{\theta'} = constant\).
So, we have the Euler-Lagrange equation for this problem (remember that \(r\)
is our variable of integration):
$$
\eqalign{
\frac{d}{dr}\pard{F}{\theta'} - \pard{F}{\theta} = 0 \cr
\text{but}\enspace \pard{F}{\theta} = 0 \cr
\text{therefore}\enspace \frac{d}{dr}\pard{F}{\theta'} = 0 \cr
\text{so}\enspace \pard{F}{\theta'} = k
}
$$
$$
\pard{}{\theta'} \left(\frac{\sqrt{1+(r\theta')^2}}{\sqrt{\frac{g}{R}(R^2-r^2)}}\right) = k
$$
Performing the differentiation we have
$$
\frac{1}{\sqrt{\frac{g}{R}(R^2-r^2)}}\cdot\frac{r^2\theta'}{\sqrt{1+(r\theta')^2}} = k
$$
Squaring both sides and solving for \((r\theta')^2\) first
$$
\frac{1}{\frac{g}{R}(R^2-r^2)}\cdot\frac{(r^2\theta')^2}{1+(r\theta')^2} = k^2
$$
$$
\begin{equation}
\frac{\frac{k^2g}{R}(R^2-r^2)}{r^2-\frac{k^2g}{R}(R^2-r^2)} = (r\theta')^2
\label{eq:rt2}
\end{equation}
$$
We solve for \((r\theta')^2\) and mark that equation because in the end, our original integral
for the minimum time is a function of \((r\theta')^2\) and we don't want to have to recompute it.
Going the final step to solve for \(\theta'\) we have
$$
\begin{equation}
\theta' = \frac{1}{r}\sqrt{\frac{\frac{k^2g}{R}(R^2-r^2)}{r^2-\frac{k^2g}{R}(R^2-r^2)}}
\label{eq:tp}
\end{equation}
$$
At this point we pause to consider the variable and limits of integration. If you look at the figure
below, we can see that to traverse the curve, \(r\) goes from \(R\) to \(R\). That doesn't help us much because the definite integral
would end up being 0.

We do notice, however, that the curve must be symmetric (the path shouldn't matter
on which side you start) and so the deepest part of the curve is in the middle at \(r_0\). At this value
of \(r=r_0\), we can see that \(\frac{dr}{d\theta} = 0\) since \(r(\theta)\) is mimimized. Conversely,
this means that \(\theta' = \frac{d\theta}{dr} \rightarrow \infty\). This happens when the denominator of equation \ref{eq:tp}
becomes zero. So, solving for the constants in this case when \(r=r_0\)
$$
r_0^2 - \frac{k^2g}{R}(R^2-r_0^2) = 0
$$
yields
$$
\frac{k^2g}{R} = \frac{r_0^2}{R^2-r_0^2}
$$
Before we find the path, we'll find the time it takes to traverse the curve. Of course, this time will be
a function of \(r_0\) so we will still have some work to do.
Rewriting equation \ref{eq:rt2} with the above substitution and then cleaning up some of the
fractions yields
$$
(r\theta')^2 = \frac{\frac{r_0^2}{R^2-r_0^2}(R^2-r^2)}{r^2-\frac{r_0^2}{R^2-r_0^2}(R^2-r^2)}
$$
yields
$$
\begin{equation}
(r\theta')^2 = \frac{r_0^2(R^2-r^2)}{r^2(R^2-r_0^2)-r_0^2(R^2-r^2)}
\label{eq:rt2r0}
\end{equation}
$$
Now, going back to our original functional we factor out the \(dr\) and put in the actual limits
of integration, multiplying the whole thing by 2 because we're only going halfway in the integral.
$$
\Delta t = 2\int_{R}^{r_0}\frac{\sqrt{1+(r\theta')^2}}{\sqrt{\frac{g}{R}(R^2-r^2)}}dr
$$
Substituting \ref{eq:rt2r0} into the above and moving some constants over to the left hand side
$$
\frac{1}{2}\sqrt{\frac{g}{R}}\Delta t = \int_{R}^{r_0}\frac{1}{\sqrt{(R^2-r^2)}}\cdot\frac{r\sqrt{R^2-r_0^2}}{\sqrt{r^2(R^2-r_0^2) - r_0^2(R^2-r^2)}}dr
$$
And after a little more algebra, factoring out the \((R^2-r_0^2)\)
$$
\frac{1}{2}\sqrt{\frac{g}{R}}\Delta t = \int_{R}^{r_0}\frac{r}{\sqrt{(R^2-r^2)(r^2 - \frac{r_0^2}{R^2-r_0^2}(R^2-r^2))}}dr
$$
Sometimes at points like this it helps to look at the units to make sure we at least have a sanity check.
On the left hand side we have \(\sqrt{\frac{m}{s^2}\cdot\frac{1}{m}}\cdot s\) which is unitless. On the right,
we have \(\frac{m\cdot m}{\sqrt{m^4}}\) which is also unitless. So we at least have that going for us.
We're almost home for this part. We now have a relatively straightforward integration. We let \(u=R^2-r^2\),
\(du = -2rdr\) and \(r^2 = R^2 - u\). These substitutions result in
$$
\frac{1}{2}\sqrt{\frac{g}{R}}\Delta t = -\frac{1}{2}\int_{R}^{r_0}\frac{1}{\sqrt{u((R^2-u) - \frac{r_0^2}{R^2-r_0^2}u)}}dr
$$
Simplifying the denominator
$$
\frac{1}{2}\sqrt{\frac{g}{R}}\Delta t = -\frac{1}{2}\int_{R}^{r_0}\frac{1}{\sqrt{u(R^2-(1+\frac{r_0^2}{R^2-r_0^2})u)}}dr
$$
A little more
$$
\frac{1}{2}\sqrt{\frac{g}{R}}\Delta t = -\frac{1}{2}\int_{R}^{r_0}\frac{1}{\sqrt{u(R^2-\frac{R^2}{R^2-r_0^2}u)}}dr
$$
A little more algebra by factoring out an \(R^2\) from the to get the integral into a more recognizable form
$$
\frac{1}{2}\sqrt{\frac{gR^2}{R}}\Delta t = -\frac{1}{2}\int_{R}^{r_0}\frac{1}{\sqrt{u(1 - \frac{1}{R^2-r_0^2}u)}}dr
$$
Simplifying the left a little and with a little help from

Wolfram-Alpha
we have (putting back the \(u=R^2-r^2\))
$$
\eqalign {
\frac{1}{2}\sqrt{gR}\Delta t &= 2\sqrt{(R^2-r_0^2)}\sin^{-1}\left(\sqrt{\frac{R^2-r^2}{R^2-r_0^2}}\right) \Bigg \bracevert_R^{r_0} \cr
\frac{1}{2}\sqrt{gR}\Delta t &= 2\sqrt{(R^2-r_0^2)}(\frac{\pi}{2} - 0) \cr
\frac{1}{2}\sqrt{gR}\Delta t &= \pi\sqrt{R^2-r_0^2}
}
$$
And finally solving for \(\Delta t\)
$$
\Delta t = 2 \pi \sqrt{\frac{R^2 - r_0^2}{gR}}
$$
Here, we can ask the question of how long will it take to go from one side of the earth to the other, straight
through the core. At that point, \(r_0 = 0\) so \(\Delta t = 2\pi\sqrt{\frac{R}{g}} \). With \(g=9.8\frac{\text{m}}{\text{s}^2}\)
and the radius of the earth \(R=6.4\cdot10^6 \text{m}\)
$$
\Delta t = 2\pi\sqrt{\frac{6.4\cdot 10^6}{9.8}} \approx 5078 \text{s} \approx 85 \text{min}
$$
Here is a graph of the transit time in minutes versus the depth as a fraction of \(R\).

Solving for the path is coming in part 3.

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