Is angular momentum conserved for a mass fixed to a horizontal guide I have the system shown below. Mass 1 confined to a vertical guide, and mass 2 confined to a horizontal guide joined together by a spring. My question is very simple: is the total angular momentum conserved with respect to the crossing point of the two guides?
This is a question set by a lecturer who says the answer is no, angular momentum is not conserved. The reasoning being that if you substitute into the Lagrangian a rotation of the coordinates then the Lagrangian is not invariant.
However, it seems to me that angular momentum is conserved, as because of the guides the two masses can only move in a radial direction so surely angular momentum is always zero and therefore conserved?
Any comments appreciated :)

 A: The fact that momentum is not conserved in this system is an indication that angular momentum is not conserved either. Let's simplify the problem a bit and consider just one mass, m2, which is constrained to move horizontally and is oscillating back and forth through the origin due to the fact that it is connected to a spring at the origin. The mass m2 goes to the left, and then a short time later is speeding toward the right, and then to the left again. Clearly, momentum is not conserved. Angular momentum is not conserved, either, except for the trivial case where you measure the angular momentum with respect to the origin, in which case it is always zero. 
Assume that you measure the angular momentum of m2 oscillating back and forth with respect to where m1 is currently shown in your diagram. Is the angular momentum of m2 with respect to this position conserved? Clearly not. When the mass m2 is speeding to the left, the angular momentum is in the clockwise direction (i.e., angular momentum vector pointing into the page), and when the mass m2 is later speeding to the right the angular momentum is in the counter-clockwise direction (i.e., angular momentum vector pointing out of the page).
So neither momentum nor angular momentum is conserved in the analogous case of a single mass oscillating back and forth due to a spring. It should be apparent that neither momentum nor angular momentum (except perhaps for some special cases such as measuring the angular momentum with respect to the origin) are conserved for the case of the two masses shown, either.
A: There exist choices of origin for which the angular momentum of the system is not conserved.
Consider, for example, an origin at $x_0\,\hat{\mathbf{x}}$, and consider the initial condition in which $m_2$ is at the origin standing still, and $m_1$ is at $L\, \hat{\mathbf y}$.  Mass $m_2$ will remain at the origin, and mass $m_1$ will oscillate so that $y_1(t) = L\cos(\omega t)$.  About the chosen origin, the angular momentum of $m_2$ is zero since it's standing still, and the angular momentum of $m_1$ is $x_0(-m\dot y_1)\hat{\mathbf z}$.  Since $\ddot y_1\neq 0$, the total angular momentum changes with time.
