On velocity-dependent force fields in which all orbits are planar

Abstract

Gonzalez-Gascon (1988) has shown that relativistic motion in a physically central force field, which is therefore planar, corresponds mathematically to motion in a non-central velocity-dependent field. His analysis is re-examined using a more compact coordinate-independent vector and tensor notation, which makes it clear that the non-central field is a special case of the form (x-c)p(x, x)+xq(x, x). It is proved that, for arbitrary scalar functions p and q, all orbits are planar in this field, which is non-central if q not=0.