# Conservation of angular momentum in scattering processes

• B
Greetings.

So... let us consider a particle moving in the yz plane, coming from the infinite towards a region were a gravitational potential is appreciable. The Lagrangian of the system is

$\mathcal{L} = \frac{1}{2}\mu (\dot{r}^2+r^2{\dot \phi}^2) + \frac{G\,m\,M}{r}$

where $\mu$ is the reduced mass and $r$ is the relative distance between the scattered particle and the particle generating the gravitational potential. From this Lagrangian we take that the quantity

$L = m\,r\,v$

is conserved, right? Now, from the angular momentum vector we have,

$\vec{L} = m\,(y\,\dot{z} - z\,\dot{y})$

and $|\vec{L}| = m\,r\,v\,\sin\theta$ where $\theta$ is the angle between $r$ and $v$.

So... my problem is, the angular momentum is then only conserved when $r$ and $v$ are orthogonal? So, it doesn't really apply to scattering, only for orbiting particles?

Thank you very much.