- #1
WWCY
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Homework Statement
I have an issue with understanding the idea of generalised momentum for the Lagrangian.
For a central force problem, the Lagrangian is given by,
$$L = \frac{1}{2}m(\dot{r} ^2 + p^2 \dot{\phi ^2}) - U(r)$$
with ##r## being radial distance.
The angular momentum is then,
$$P_{\phi} = \frac{\partial L}{\partial \dot{\phi}} = mr^2\dot{\phi}$$
Without the Lagrangian, I would have started with,
$$P_{\phi} = \vec{r} \times \vec{P}$$
$$|P_{\phi}| = rP\sin (\theta)$$
$$|P_{\phi}| = rmv \sin (\theta)$$
with ##\theta## being the angle between the two vectors, which I think can be reduced with the equation
$$v = r\dot{\phi}$$
and angular momentum is,
$$|P_{\phi}| = mr^2 \dot{\phi} \sin (\theta)$$
which varies significantly.
Could someone kindly
a) Explain the errors in my assumptions
b) Explain how I should have altered my "Newtonian" working in order to derive the same result as the Lagrangian method?
Thanks!