Lagrangian of particle moving on a sphere

[Pyroadept]

Homework Statement

Find the shape of a tunnel drilled through the Moon such that the travel time between two points on the surface of the Moon under the force of gravity is minimized. Assume the Moon is spherical and homogeneous.

Hint: Prove that the shape is the hypercycloid

x(θ) = (R - r)cos[(R/r)θ] + rcos[{(R-r)/R}θ]
y(θ) = (R - r)sin[(R/r)θ] - rsin[{(R-r)/R}θ]

Homework Equations

L = T - U

Euler-Lagrangian equation: ∂L/∂q = (d/dt)(∂L/∂{dq/dt})

The Attempt at a Solution

Hi, here's what I've done so far:

L = T - U
= m/2(v_x^2 + v_y^2 + v_z^2) - mgz

Suppose the moon is a sphere centered at the point a on the z (vertical) axis. Suppose particle is positioned at top of moon (i.e. (0,0,2a)

Then L = m/2(v_x^2 + v_y^2 + v_z^2) - 2mga

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But where do I go from here? Is this even correct? The hint seems to suggest I should be converting to polar coordinates at some point, but how can I do this when I don't know what z is? How do I bring the travel-in-least-time aspect into it?

Please point me in the right direction!

Thanks for any help.

 

[lanedance]

I haven't attempted it, but notice polar coordinates are the natural choice, as gravity acts radially

so you need to start with 2 points on the surface of the moon, separated say by an angle

You can bring the travel time into it by setting up an integral that computes the travel time for an arbitrary path, say $f = r(\theta)$. The integrand will be the function you use in your Euler Lagrange equations, to find the function f which minimises the integral