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A potential given by : $$ V(x,y,z) = \frac{1}{2}m(x^2+y^2+\frac{z^2}{2}). $$ Which component of angular momentum is conserved.

An attempt:

Angular momentum along z, $ L_{z} = m(x\dot{y} - y\dot{x})$

and $ \frac{dL_{z}}{dt} = m(\dot{x}\dot{y}+x\ddot{y}-\dot{y}\dot{x}-\ddot{x}y) = m(x\ddot{y}-\ddot{x}y)$

But from equations of motion gives us

$ \ddot{x} = -\frac{k}{m}x $

$ \ddot{y} = -\frac{k}{m}y $

using this,

$ \frac{dL_{z}}{dt} = -kxy + kxy = 0 $

So, is $L_{z}$ conserved ? I know we can also try to work out the poisson bracket method, but this seemed slightly easier.

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up vote 1 down vote accepted

You are right. However a shorter answer is the following: $V$ and thus the whole Lagrangian is invariant under rotations around the axis $z$. Noether theorem immediately implies that the $z$ component of the angular momentum is conserved.

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That is right, how silly I didn't think about it !! Thanks a lot. – user35952 Jan 7 '14 at 16:33

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