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?
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 [tex]\theta[/tex] or [tex]\phi[/tex] and their derivatives for your q and q dot things, but i guess its possible. Here is my best stab, and im pretty sure of the strength of this statement: Anytime there is a conservation law, it means there is a symmetry in any one 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...