= Zero-energy circular motion in a point-dipole field
{title2=$d^2r^2/dt^2=4E/m$}
For <potential energy> homogeneous of degree $-2$, <Newton's second law> gives $d^2r^2/dt^2=4E/m$. Indeed, Euler's identity implies $\mathbf r\cdot\mathbf F=2U$, so the second derivative equals $2v^2+4U/m=4(K+U)/m$. Thus a trajectory with zero total <energy> and initially zero radial velocity remains on a sphere of constant radius. A positive <electric charge> released from rest on the equator of a fixed ideal <electric dipole> has these properties; in its meridional plane it oscillates on the semicircle on the negative-pole side. A finite two-charge <electric dipole> does not have exactly homogeneous <potential energy> and perturbs this result.
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