A wire is bent into the shape of a quartic function \(y=a x^{4}\) and oriented in a vertical plane, with \(x\) horizontal, \(y\) vertical, and \(a\) a positive constant. A bead of mass \(m\) is threaded onto the wire, and the wire is then forced to rotate with constant angular velocity \(\Omega\) about the \(y\) axis.

(a) Let \(x\) be the generalized coordinate for the bead and find its Lagrangian.

(b) Is the bead's energy conserved? Why or why not?

(c) Is the bead's angular momentum conserved about the vertical axis? Why or why not?

(d) Find the bead's Hamiltonian. Is \(H\) conserved? Why or why not?

(e) Is \(E=H\) ? Why or why not?

(f) Given \(\Omega>0\), are there any equilibrium positions of the bead? (g) If so, Is each stable or unstable? For any stable equilibrium position, find the frequency \(\omega\) of small oscillations about the equilibrium point, expressed as a multiple of \(\Omega\).