Planetary motion in four and higher dimensions.
Consider the motion of planets in planar circular orbits around heavy stars in our four dimensional spacetime and in spacetimes with additional spatial dimensions. We wish to study the stability of these orbits under perturbations that keep them planar. Such a perturbation would arise, for example, if a meteorite moving on the plane of the orbit hits the planet and changes its angular momentum.
Show that while planetary circular orbits in our four-dimensional world are stable under such perturbations, they are not so in five or higher dimensions. [Hint: You may find it useful to use the effective potential for motion in a central force field.]