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July  2021, 41(7): 3447-3464. doi: 10.3934/dcds.2021003

## Constant-speed ramps for a central force field

 1 Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain 2 Department of Mathematics, Central Connecticut State University, New Britain, CT 06050, USA

* Corresponding author

Received  March 2020 Published  July 2021 Early access  January 2021

Fund Project: Rafael López has partially supported by the grant no. MTM2017-89677-P, MINECO/AEI/FEDER, UE

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. Not only they are solutions but every other solution approaches either a circle or a logarithmic spiral.

Citation: Rafael López, Óscar Perdomo. Constant-speed ramps for a central force field. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3447-3464. doi: 10.3934/dcds.2021003
##### References:
 [1] J. Bertrand, Théorème relatif au mouvement d'un point attiré vers un centre fixe, C. R. Acad. Sci., 77 (1873), 849-853.   Google Scholar [2] H. Goldstein, Classical Mechanics, Addison-Wesley, 2nd. edition, Reading, MA, 1980.  Google Scholar [3] O. M. Perdomo, A dynamical interpretation of cmc Twizzlers surfaces, Pacific J. Math., 258 (2012), 459-485.  doi: 10.2140/pjm.2012.258.459.  Google Scholar [4] O. M. Perdomo, Helicoidal minimal surfaces in $\mathbb{R}^3$, Illinois J. Math., 57 (2013), 87-104.  doi: 10.1215/ijm/1403534487.  Google Scholar [5] O. M. Perdomo, Constant-speed ramps, Pacific J. Math., 275 (2015), 1-18.  doi: 10.2140/pjm.2015.275.1.  Google Scholar [6] A. P. Usher, A History of Mechanical Inventions, Revised Edition. Dover, 1989.  Google Scholar [7] Wolfram Mathematica 7 Documentation. Google Scholar

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##### References:
 [1] J. Bertrand, Théorème relatif au mouvement d'un point attiré vers un centre fixe, C. R. Acad. Sci., 77 (1873), 849-853.   Google Scholar [2] H. Goldstein, Classical Mechanics, Addison-Wesley, 2nd. edition, Reading, MA, 1980.  Google Scholar [3] O. M. Perdomo, A dynamical interpretation of cmc Twizzlers surfaces, Pacific J. Math., 258 (2012), 459-485.  doi: 10.2140/pjm.2012.258.459.  Google Scholar [4] O. M. Perdomo, Helicoidal minimal surfaces in $\mathbb{R}^3$, Illinois J. Math., 57 (2013), 87-104.  doi: 10.1215/ijm/1403534487.  Google Scholar [5] O. M. Perdomo, Constant-speed ramps, Pacific J. Math., 275 (2015), 1-18.  doi: 10.2140/pjm.2015.275.1.  Google Scholar [6] A. P. Usher, A History of Mechanical Inventions, Revised Edition. Dover, 1989.  Google Scholar [7] Wolfram Mathematica 7 Documentation. Google Scholar
A mass $M$ sliding along $\alpha$ under the effect of a central force ${\mathbf F}({\mathbf r})$ and the friction force
Case $v = 1$. The phase portrait of the system (17), with $\mu = 0.5$. The origin $(0,0)$ is the only equilibrium point and it is a stable focus
Case $v = 1$. The purple part of the logarithmic spiral (left) is the TreadmillSled of the non-circular ramp (right)
The phase portrait of the system (21). Left: $v = 2$ and $\mu = 0.1$. Right: $v = 0.5$ and $\mu = 0.3$
TreadmillSleds that are half-lines. Left: $v = 2$, $\mu = 0.1$. Right: $v = 0.5$, $\mu = 0.3$
, left. Here $v = 2$, $\mu = 0.1$. Left: parametrization (23) for $\mathbf{a}$. Right: parametrization (23) for $-\mathbf{a}$">Figure 6.  Constant-speed ramps whose TreadmillSleds are half-lines of Figure 5, left. Here $v = 2$, $\mu = 0.1$. Left: parametrization (23) for $\mathbf{a}$. Right: parametrization (23) for $-\mathbf{a}$
, right. Here $v = 0.5$ and $\mu = 0.3$. Left: parametrization (23) for $\mathbf{a}$. Right: parametrization (23) for $-\mathbf{a}$">Figure 7.  Constant-speed ramps $\alpha_{ls}$ whose TreadmillSleds $\gamma_{ls}$ are the half-lines of Figure 5, right. Here $v = 0.5$ and $\mu = 0.3$. Left: parametrization (23) for $\mathbf{a}$. Right: parametrization (23) for $-\mathbf{a}$
Constant-speed ramps whose TreadmillSleds are not half-lines. Here $v = 2$ and $\mu = 0.1$. Left: the TreadmillSled which is asymptotic to the line of vector $\mathbf{a}$. Right: the constant-speed ramp
Constant-speed ramps whose TreadmillSleds are not half-lines. Here $v = 0.8$ and $\mu = 0.3$. Left: the TreadmillSled. Right: the constant-speed ramp
Constant-speed ramps that are not spirals. Left: $v = 2$ and $\mu = 0.1$. Right: $v = 0.8$ and $\mu = 0.3$
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