Computing the action for a particle in a gravitational field

  • #1

Main Question or Discussion Point

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

[tex]
S = \int_{t_i}^{t_f} L(q,\dot q,t) dt,
[/tex]

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

[tex]
L = \frac 12 m \dot{\mathbf{r}}^2 + \frac{\alpha}{r}
[/tex]

(where, of course, [itex]|\mathbf{r}| = r[/itex]) so that

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

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

Answers and Replies

  • #2
Wow...well, I guess I don't feel so bad about not having been able to do this now!
 
  • #3
dextercioby
Science Advisor
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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
 

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