Repeat the preceding problem for a particle sliding inside a smooth cone defined by \(z=\alpha r\).

Data from preceding problem

A particle of mass \(m\) slides inside a smooth paraboloid of revolution whose axis of symmetry \(z\) is vertical. The surface is defined by the equation \(z=\alpha ho^{2}\), where \(z\) and \(ho\) are cylindrical coordinates, and \(\alpha\) is a constant. There is a uniform gravitational field \(g\).

(a) Select two generalized coordinates for \(m\).

(b) Find \(T, U\), and \(L\).

(c) Identify any ignorable coordinates, and any conserved quantities.

(d) Show that there are two first integrals of motion, and find the corresponding equations.