Optimizing Tunnel Design: Calculus of Variations on a Spherical Earth

[No Name Required]

Presume the Earth is spherical, homogeneous and of radius $$R$$. What should be the shape of a tunnel connecting two points on the surface in order to minimize the time it takes for a particle to travel between the two points.

I have had a go at doing it in both polar and cartesian co-ordinates but am getting stuck. I'm fairly sure we are supposed to do it in cartesian but this way is proving particularly tricky.

What i have done is this;

Using some basic physics and energy conservation I have found that

$$\displaystyle{v(r) = \sqrt{\frac{g}{R}(R^2 - r^2)}}$$

or

$$\displaystyle{v = \sqrt{\frac{g}{R}(R^2 - x^2 - y^2)}}$$

$$ds^2 = dx^2 + dy^2$$

$$ds = \sqrt{1 + (y')^2} \; dx$$

So $$\displaystyle{\int t = \frac{ds}{v} = \sqrt{\frac{R}{g}} \int \frac{\sqrt{1 + (y')^2}}{\sqrt{R^2 - x^2 - y^2} \; dx}$$

The functional is

$$\displaystyle{T[x, y, y'] = \sqrt{\frac{R}{g}} \frac{\sqrt{1 + (y')^2}}{\sqrt{R^2 - x^2 - y^2}}$$

This is the problem, I've never dealt with a functional like this before. Up until now, any functional has had a cyclic coordinate which has made it nice and simple. This fuctional depends explicitly on both $$x$$ and $$y$$.

Is what I've done so far correct? How can I go about finishing the problem


The question also gives the answer that should be obtained

$$\displaystyle{x(\theta) = (R - r) \cos \left(\frac{R}{r}\theta \right) + r \cos \left(\frac{R - r}{R} \theta \right)}$$

and

$$\displaystyle{y(\theta) = (R - r) \sin \left(\frac{R}{r}\theta \right) - r \sin \left(\frac{R - r}{R} \theta \right)}$$

 

[No Name Required]

bump... any ideas anyone?

 

[BerkMath]

Assuming your functional is correct, why don't you proceed with calculating the Euler-Lagrange differential equation. If you do it in polar coordiantes with the correct functional, the ELDE can be integrated once giving the equation for a hypocycloid in polar coordiantes. It can then be written parametrically giving you the answer your book provided. I don't know why your professor would have a bias towards either method, generally they like to see the simplest solution.

 

[CarlB]

So you get to neglect Coriolis effects?

Carl

 

[BerkMath]

I found I a cool link on a GR brachistrone problem. It becomes an interesting problem in that there are then 2 distinct frames from which to minimize the travel time.