Scalar radial formulation of the rear-track differential equation

Determine whether the rear-track differential equation R′(t) = F′(t) · (F(t) − R(t)) [F(t) − R(t)] can be reformulated for spiral tracks as a single scalar differential equation directly for the radial function r(t) with polar angle parameter t; either construct such an equation or show that it cannot be done under the spiral assumptions.

Background

The standard rear-track equation is vector-valued and depends on both the front position F(t) and the rear position R(t). Because the target limit curve is spiral (radius depending on angle), the authors consider whether this structure permits a reduction to an equation solely for the radius r(t).

They explicitly state their inability to find such a reduction, highlighting a concrete unresolved formulation question.