### Planetary motion in four and higher dimensions

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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.]

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