The problem of interest is worded as follows:
A horizontal square wire frame with vertices $ABCD$ and side length $2a$ rotates with constant angular frequency $\omega$ about a vertical axis through $A$. A bead of mass $m$ is threaded on $BC$ and moves without friction. The bead is connected to $B$ and $C$ by two identical light springs of force constant $k$ and equilibrium length $a$.
By considering a small displacement of the particle away from the midpoint of $BC$, find the Lagrangian.
I'm struggling to find a good coordinate system to work in. I considered taking A to be the origin, and using a polar angle measured from the line produced by the initial position of $AM$ where $M$ is the midpoint of $BC$, but the expression for the angle that the particle makes with the initial line seems far too complicated to be the best way.
Any hints on how to proceed?