Lagrangian of a sliding ladder

1. The problem statement, all variables and given/known data
A ladder of length 2l and mass m leans against a smooth wall and rests on a smooth floor. The ladder initially makes an angle θ0 to the vertical. It slides downwards maintaining contact with both the wall and the floor. Calcula the the Lagrangian and the conjugate momentum, and find the equation of motion. (The moment of inertia of a rod of length a and mass M about an axis through its centre perpendicular to the rod is Ma2/12.)

3. The attempt at a solution
I have L=(2/3)ml2(dθ/dt)2-mglcosθ with θ the time varying angle to the vertical. I’m confident this is correct as a similar result is obtained here for a slightly different definition of θ

Therefore why is the question asking for a conjugate momentum?

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