Constant-speed ramps for a central force field

TL;DR

This paper characterizes all planar ramps where a particle moves at constant speed under a specific central force, revealing that solutions are either circles or logarithmic spirals, with other solutions approaching these forms.

Contribution

It provides a complete description of constant-speed ramps under the force F(r) = -m/r, highlighting the roles of circles and logarithmic spirals as fundamental solutions.

Findings

Solutions include circles and logarithmic spirals.

Other solutions asymptotically approach these fundamental curves.

The analysis applies specifically to the force F(r) = -m/r.

Abstract

We investigate the problem of determining the planar curves that describe ramps where a particle of mass $m$ moves with constant-speed when is subject to the action of the friction force and a force whose magnitude $F(r)$ depends only on the distance $r$ from the origin. In this paper we describe all the constant-speed ramps for the case $F(r)=-m/r$. We show the circles and the logarithmic spirals play an important role. No only they are solutions but every other solution approaches either a circle or a logarithmic spiral.