Tangentially driven motion on an axisymmetric surface (${\dot{\varphi }}^2=\frac{g{h}^{\prime }(r)}{r+2r_{\varphi }^2/r-r_{\varphi \varphi }}$)

Neglecting vertical acceleration and the vertical driving-force component, the radial normal reaction is $-Mg{h}^{\prime }(r)$. A drive parallel to the horizontal velocity cancels when the radial acceleration equation is combined with $r_{\varphi }/r$ times the azimuthal equation. For a prescribed path $r(\varphi )$ this gives the displayed speed constraint. It describes the projected approximation, not arbitrary trajectories with substantial vertical motion.

// Target: classical-mechanics.bigb