Say you were to travel from Paris to Tokyo by digging a tunnel between both cities. If the tunnel is straight, one can easily compute that the time for travelling from one city to the other (independently from where both cities are actually) is a constant (about 42 minutes). It goes without saying : only gravitationnal potential and no friction.
But let's upgrade on the idea. Fix the two cities : what is the optimal shape of the tunnel, so that the time it takes for going from one city to the other is shortest ?
I assume a planar movement, where the plane contains both cities and the center of the earth. With Gauss' law, it is easy to compute that the gravitationnal potential on a mass m inside the earth at any distance $r$ from the center :
$V(r) = m\alpha r^2$ where $\alpha$ is a constant. I parameterize the tunnel by a polar function $r(\theta)$ where the origin is taken at Earth's center. The problem reduces to minimizing the following functional
$\mathcal{T}(\dot{r},r,\theta) = \int \limits_{\theta_1}^{\theta_2} \frac{ds}{|v|} = \int \limits_{\theta_1}^{\theta_2} d\theta \frac{\sqrt{\dot{r}^2 + r^2}}{|v|} = \sqrt{\frac{m}{2}}\int \limits_{\theta_1}^{\theta_2} d\theta \frac{\sqrt{\dot{r}^2 + r^2}}{\sqrt{E - m\alpha r^2}} \equiv \sqrt{\frac{m}{2}}\int \limits_{\theta_1}^{\theta_2} d\theta G(\dot{r},r,\theta)$
$\theta_1$ and $\theta_2$ are the angular coordinates of both cities. $\dot{r}$ is the derivative of $r$ with respect to $\theta$. With this parametrisation, the 'Hamiltonian' is conserved :
$\frac{\partial G}{\partial \dot{r}}\dot{r} - G = c$
Which leads to
$\frac{-r^2}{\sqrt{E-2\alpha r^2}\sqrt{\dot{r}^2+r^2}} = c$
And here I get stuck. Even if I try to play with special initial conditions, I cannot solve this equation. The easiest way is to perform separation of variables, but the integralcannot be carried out.
Any ideas on how to solve this problem ? I am sure the solution is a nice well known symmetric function. I just don't know how to tackle this problem. Would you tink it'd be easier to solve this in cartesian coordinates ?
