A wire is bent into the shape of a cycloid, defined by the parametric equations \(x=A(\varphi+\sin \varphi)\) and \(y=A(1-\cos \varphi)\), where \(\varphi\) is the parameter \((-\pi<\varphi<\pi)\), and \(A\) is a constant. The wire is in a vertical plane, and is spun at constant angular velocity \(\omega\) about a vertical axis through its center. A bead of mass \(m\) is slipped onto the wire.

(a) Find the Lagrangian of the bead, using the parameter \(\varphi\) as the generalized coordinate.

(b) Identify any first integral of motion of the bead.

(c) Reduce the problem to quadrature: That is, show that the time \(t\) can be expressed as an integral over \(\varphi\).