cmp0xff
commented
6 months ago Landau vol 1, sec. 14
If we reduce to 2d, $$L = \frac{m}{2} \left(\dot{r}^2 + r^2 \dot\phi^2\right) - V(r)\,.$$
$L$ does not contain $\phi$ explicitly, making $\phi$ a cyclic variable. We have an integral of motion $l \equiv m r^2 \phi$.
$L$ does not contain $t$ explicitly. We have a second integral of motion, the energy $$E = \frac{m}{2}\left(\dot r^2 + r^2 \dot\phi^2\right) + V(r) = \frac{m}{2} \dot{r}^2 + \frac{l}{2mr^2} + V(r)\,.$$ We can derive $$\frac{\mathbb{d}t}{\mathbb{d}r} = \left(\frac{2}{m}\left(E-V(r)\right) - \frac{l^2}{m^2 r^2}\right)^{-1/2}\,.$$
emptymalei
Definition
A generic central force problem.
References