Computing the action for a particle in a gravitational field

[AxiomOfChoice]

A friend of mine told me he fielded this at an oral exam: "Compute the classical action S for a particle of mass m in a gravitational field U = -\alpha/r." I know the formula for the classical action is given by

<br /> S = \int_{t_i}^{t_f} L(q,\dot q,t) dt,<br />

and that for a particle in a gravitational field, we have

<br /> L = \frac 12 m \dot{\mathbf{r}}^2 + \frac{\alpha}{r}<br />

(where, of course, |\mathbf{r}| = r) so that

<br /> S = \int_{t_i}^{t_f} \left( \frac 12 m \dot{\mathbf{r}(t)}^2 + \frac{\alpha}{r(t)} \right) dt.<br />

But how in the WORLD am I supposed to perform this integration? Am I supposed to derive expressions for \mathbf{r} and r as functions of t?

 

[AxiomOfChoice]

Wow...well, I guess I don't feel so bad about not having been able to do this now!

 

[dextercioby]

You don't know who r(t) is, so you can't compute the integral, can you ? I think your last expression is exactly what the problem/examinator asked for. So A^+, huh ? :D