Is Angular Momentum Conserved in a Central and Gravitational Force System?

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SUMMARY

The discussion focuses on determining the conservation of angular momentum in a system influenced by both central and gravitational forces. The participant has calculated the Lagrangian and equations of motion but seeks clarification on checking for angular momentum conservation. Key insights include the importance of the invariance of the system under rotation and the relationship between the conjugate momentum and angular momentum. Additionally, it is essential to compute the Hamiltonian (H) and verify if it commutes with the angular momentum operator (J) to establish conservation.

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I have a problem with a particle experiencing a central force towards some origin, as well as a gravitational force downwards. I've calculated the Lagrangian, and the equations of motion. Now I'm being asked to see if the system follows conservation of angular momentum. How do I do this? I know it has something to do with seeing if the system is invariant of rotation, but how do I check for that?
 
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If \frac{\partial L}{\partial q} =0 then the conjugate momentum \frac{\partial L}{\partial \dot{q}} is a conserved quantity.

If that doesn't clear things up, then post what you have for the Lagrangian.
 
But the conjugate momentum is the same as the angular momentum only in some cases.
Compute H and chech if H commutes with J.
 

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