Constant-speed ramps for a central force field

Abstract

<p style='text-indent:20px;'>We investigate the problem of determining the planar curves that describe ramps where a particle of mass <inline-formula><tex-math id="M1">\begin{document}$ m $\end{document}</tex-math></inline-formula> moves with constant-speed when is subject to the action of the friction force and a force whose magnitude <inline-formula><tex-math id="M2">\begin{document}$ F(r) $\end{document}</tex-math></inline-formula> depends only on the distance <inline-formula><tex-math id="M3">\begin{document}$ r $\end{document}</tex-math></inline-formula> from the origin. In this paper we describe all the constant-speed ramps for the case <inline-formula><tex-math id="M4">\begin{document}$ F(r) = -m/r $\end{document}</tex-math></inline-formula>. We show the circles and the logarithmic spirals play an important role. Not only they are solutions but every other solution approaches either a circle or a logarithmic spiral.