Rolling constraint inside a moving circular track ($r\dot{\psi }=R\dot{\theta }-(R-r)\dot{\varphi }$)

A circular body of radius $r$ rolls inside a circular track of radius $R$. Let $\theta$ be the track's material rotation, $\varphi$ the absolute angle of the line joining the two centres, and $\psi$ the inner body's absolute spin angle, all with one consistent orientation. Equating contact-point velocities gives the displayed constraint. Translation of the track's centre cancels from that equality. A stationary track gives $\dot{\psi }=-(R-r)\dot{\varphi }/r$.