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

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In a system with a particle experiencing both a central force and gravitational force, conservation of angular momentum can be assessed by examining the invariance of the Lagrangian under rotations. The condition \(\frac{\partial L}{\partial q} = 0\) indicates that the conjugate momentum \(\frac{\partial L}{\partial \dot{q}}\) is conserved. To further analyze the system, one should compute the Hamiltonian \(H\) and check if it commutes with the angular momentum operator \(J\). This approach helps confirm whether angular momentum is conserved in the given force system. Understanding these relationships is crucial for determining the conservation laws in mechanics.
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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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