Lagrangians and conserved quantities

[quasar_4]

Hi,

I have a relatively straight forward question. If we have a Lagrangian that only depends on time and the position coordinate (and its derivative), how can I decide whether angular momentum is conserved?

That is, if the Lagrangian specifically does not have theta or phi dependence, does that mean that angular momentum is always conserved?

 

[jfy4]

I think this is a really good question! I haven't thought of this before until now.

I imagine it would be a little weird for you to be interested in a situation with angular symmetry and not using $$\theta$$ or $$\phi$$ and their derivatives for your q and q dot things, but i guess its possible.

Here is my best stab, and I am pretty sure of the strength of this statement: Anytime there is a conservation law, it means there is a symmetry in anyone of the 4 spatial coordinates. Conservation laws are geometrically based, so look at your system, and if there is a symmetry in one of the coordinates, then there is conservation of something.


I hope this helps...