A particle of mass \(m\) slides on the inside of a frictionless vertically-oriented cone of semi-vertical angle \(\alpha\).

(a) Find the Hamiltonian \(H\) of the particle, using generalized coordinates \(r\), the distance of the particle from the vertex of the cone, and \(\varphi\), the azimuthal angle.

(b) Write down two first-integrals of motion, and identify their physical meaning.

(c) Show that a stable circular (constant - \(r\) ) orbit is possible, and find its value of \(r\) for given angular momentum \(p_{\varphi}\).

(d) Find the frequency of small oscillations \(\omega_{\text {osc }}\) about this circular motion, and compare it with the frequency of rotation \(\omega_{\text {circle }}\).

(e) Is there a value of the tilt-angle \(\alpha\) for which the two frequencies are equal? What is the physical significance of the equality?