A particle P of mass m slides on the smooth inner surface of a circular cone of semi-angle α. The axis of symmetry of the cone is vertical with the vertex O pointing downwards. Take as generalised coordinates r, the distance OP, and ϕ, the azimuthal angle about the vertical through O. Obtain Lagrange's equations. Show that ϕ is a cyclic coordinate and find (and identify) the conserved momentum pϕ.