# Sphere with Tunnel

1. Feb 4, 2009

### nfm1337

1. The problem statement, all variables and given/known data
A train slides without friction inside a tunnel drilled through Earth from
station A to station B. Find the curve the tunnel should follow, in order for the duration of travelling from A to B to be as short as possible. Assume Earth to be a (non-rotating) homogeneous sphere.

2. Relevant equations
Transit time:
$$T_{AB} = \int_A^B \frac{ds}{V}$$

Euler-Lagrange:
$$\frac{\delta f}{\delta y} - \frac{d}{dx} \frac{\delta f}{\delta f'} = 0$$

The gravitational acceleration at a distance r from the centre of the sphere, if r < the radius of the sphere:
$$g = \frac{4}{3} \pi \rho G r$$

3. The attempt at a solution
I tried to find an expression for the velocity, v. First by using the law of conservation of energy $$1/2 mv^2 = mgh$$, but then when I plugged this expression into the integral for the transit time, and then plugged the expression to be integrated into the Euler-Lagrange equation, I ended up with differential equations which seemed impossible to me. Then by trying to find an expression for the acceleration, as the scalar product of the gravitational acceleration and the unit tangent vector of the curve. But again, I ended up with horrible differential equations.

Then I checked on MathWorld (http://mathworld.wolfram.com/SpherewithTunnel.html) but I couldn't even understand how they reached the first equation.

I would be very happy if someone could help me with this problem. I've spent all day on it and I feel like an idiot.

Last edited: Feb 5, 2009