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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)}
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)}