# Why Goldstein's book is claiming that radius and angle doesn't contain time variable even there is $\dot{r}$ and $\dot{\theta}$?

$$L=\frac{1}{2}m(\dot{r}^2+r^2\dot{\theta}^2)-V(r)$$ $$p_\theta=\frac{\partial L}{\partial \theta}=mr^2\dot{\theta}$$ $$\dot{p}_\theta=\frac{d}{dt}(mr^2\dot{\theta})$$

Goldstein wrote that $$\dot{P}_\theta=0$$. I know $$r$$ and $$\theta$$ both (function) have time variable. Although why he wrote differentiation of momentum respect to time is 0?

• Hello! It is preferable to type out screenshots or images of text; for formulae, one can use MathJax. Thanks! Sep 9, 2021 at 12:52
• That isn't what is stated. The angular momentum is conserved, so there is no time variation of the angular momentum. Sep 9, 2021 at 12:54

You are correct that $$r$$ and $$\theta$$ are functions of time, and Goldstein is not claiming that they arent. The whole point of the discussion is that $$p_{\theta}$$ is a constant, even though it may not appear so at first since $$r = r(t)$$ and $$\theta = \theta(t)$$. This follows from the Euler Lagrange equations, there is one for each generalized coordinate $$\frac{d}{dt} \frac{\partial \mathcal L}{\partial \dot \theta} = \frac{\partial \mathcal L}{\partial \theta}, \qquad \frac{d}{dt} \frac{\partial \mathcal L}{\partial \dot r} = \frac{\partial \mathcal L}{\partial r}.$$ The first one implies $$\frac{d}{dt} \underbrace{\frac{\partial \mathcal L}{\partial \dot \theta}}_{\equiv p_{\theta}} = \frac{\partial \mathcal L}{\partial \theta} = 0.$$