- #1
AxiomOfChoice
- 533
- 1
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]?
[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]?