91Ó°ÊÓ

Two particles, each of mass \(m\), are moving under their mutual gravitational attraction, which is given by the potential \(U=-\gamma m / 2 r\), where \(2 r\) is their separation and \(\gamma\) is a constant. Find the equations of motion in terms of the coordinates \(X, Y, Z, r, \theta, \varphi\), where \(X, Y\), and \(Z\) are the Cartesian coordinates of the centre of mass and \(r, \theta\), and \(\varphi\) are the polar coordinates of one particle relative to the centre of mass.

A particle of mass \(m\) is constrained to move under gravity on the surface of a smooth right circular cone of semi-vertical angle \(\pi / 4\). The axis of the cone is vertical, with the vertex downwards. Find the equations of motion in terms of \(z\) (the height above the vertex) and \(\theta\) (the angular coordinate around the circular cross-sections). Show that $$ \dot{z}^{2}+\frac{h^{2}}{2 z^{2}}+g z=E $$ where \(E\) and \(h\) are constant. Sketch and interpret the trajectories in the \(z, \dot{z}\)-plane for a fixed value of \(h\).