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Prove: angular momentum is preserved

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Problem Statement
A particle of mass ##m## is moving in a central field with potential ##V(r)## the lagrangian in Spherical coordinate is ##l=\frac{1}{2}m(\dot{r}^2+r^2\dot{\theta}^2+r^2sin^2\theta\dot{\phi}^2)-V(r)##
Relevant Equations
##P_i=\frac{\partial }{(\partial \dot{p}_i )}##

##H(p,q)=\sum_(i=1)^n(p_i\cdot \dot{r}_i)-L##

##\dot{q}=\frac{\partial H}{(\partial p_i )}##

##\dot{p}_i=\frac{-\partial H}{(\partial q_i }##
3. Find the hamilton equations
4. using 3. prove the the angular momentum in the z axis ##L_z=m(x\dot y-xy\dot)## is preserved.

I got in ##3##:

245209


How can I prove 4?
 

PeroK

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The key equation is ##\dot{P_{\phi}} = 0## as that represents a conservation law.

You also have an equation for ##P_{\phi}## there. Can you show that is equal to ##L_z##?
 
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The key equation is ##\dot{P_{\phi}} = 0## as that represents a conservation law.

You also have an equation for ##P_{\phi}## there. Can you show that is equal to ##L_z##?
Do you mean
##-\frac{\partial H}{\partial \phi}=m(x\dot{y}-y\dot{x})##

Where ##x=rsin\theta cos\phi, y=rsin\theta sin\phi##?
 

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