In Goldstein Book it is given that:
Since the problem is spherically symmetric, the total angular momentum vector $$\boldsymbol{L}=\boldsymbol{r}\times \boldsymbol{p}$$ is conserved.
What does the quote mean and how to prove it?
Classical Mechanics - Goldstein (3rd ed) (Sec 3.2)
A spherical symmetric potential means, the potential $V$ depends only on the radius $r$, but not on the direction of vector $\boldsymbol{r}$. Hence $$V = V(r).$$ From that you get the force $$\boldsymbol{F} = -\boldsymbol{\nabla} V(r) = -\frac{\mathrm d V(r)}{\mathrm d r} \hat{\boldsymbol{r}},$$ which means that the force is parallel to the unit-vector $\hat{\boldsymbol{r}}$, and thus parallel to $\boldsymbol{r}$.
Using this together with $\dot{\boldsymbol{p}}=\boldsymbol{F}$ and $\dot{\boldsymbol{r}} = \frac{\boldsymbol{p}}{m}$, you can straight-forward calculate $\dot{\boldsymbol{L}}$ (the time-derivative of the angular momentum $\boldsymbol{L}=\boldsymbol{r}\times \boldsymbol{p}$) and proove it to be $\boldsymbol{0}$. This means that angular momentum $\boldsymbol{L}$ doesn't vary with time, i.e. it is conserved.
