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## 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]?

[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]?