What is non geodesic motion?
Obvious definition- motion that does not follow a geodesic! In relativity, geodesics are the paths followed by objects with no "outside" forces acting on them (gravity not being considered an "outside" force here) so non-geodesic motion is motion with no outside forces.
Just to elaborate quickly on what HallsofIvy said: in general relativity, the world line of a (nonspinning) test particle is a geodesic, i.e. a curve with zero path curvature. More generally, the path curvature at any event on some world line is just the magnitude of acceleration experienced by the corresponding particle. For example, a charged particle in an "electrovacuum solution" of the Einstein field equation (EFE) will experience a nonzero Lorentz force, whose magnitude (at each event on the world line of the particle) agrees with the path curvature.
I'm guessing you meant either "geodesic motion is motion with no outside forces" or else "non-geodesic motion is motion with outside forces"?
Good catch, JesseM, I missed that! But I am sure HallsofIvy simply mistyped.
Why spinning particles don't have a geodesic path?
Spin-spin forces plus an essential PF link
I didn't quite say that. However, in gtr, a spinning object immersed in a nonzero gravitational field (perhaps caused by a much larger nearby and spinning object) will in general experience a tiny "spin-spin" force and thus will be pushed off a geodesic path. However, these forces would be much too small to measure in any currently envisioned solar system experiments (at least, not any I know of). See for example "Dixon-Papapetrou equations" in Stephani, General relativity.
By the way, I urge all members to carefully read https://www.physicsforums.com/showthread.php?t=5374 if they have not already done so, and indeed to bookmark it for future reference.
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