A person on a carousel can be considered as a point mass m on a homogeneous horizontal disc with radius a and mass M. The disc rotates about a vertical axis through its centre with no friction.

The person clings to a straight railing which extends from the centre of the disc to its perimeter.

The person's distance from the centre is a function of time, R(t) and the angle f between the East and the railing is a dynamical variable.

I have to find the Lagrangian for the system and deduce that p(f) is conserved, i.e that the angular momentum is conserved.

My attempt: I was thinking that the system has no potential energy since we are talking about a horizontal disc. The system has only kinetic energy, K=K(0)+K(rel) where K(0) is due to the centre of mass motion and K(rel) is due to the motion of the particle.

Now, K(rel) = 0.5 * (total moment of inertia of the system) * (f') ^2
The total moment of inertia = 0.5 * M * a^2 [for the disc] + m * R(t)^2 [for the person]

So K(rel) = 0.5 * ( 0.5 * M * a^2 + m * R(t)^2) * (f')^2

To find K(0) we have to find some constraints for the system and this is the point I am having difficulties.

I don't know how to proceed finding the Lagrangian, any help would be greatly appreciated!!!