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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 |
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.
References:
[1] |
J. Bertrand,
Théorème relatif au mouvement d'un point attiré vers un centre fixe, C. R. Acad. Sci., 77 (1873), 849-853.
|
[2] |
H. Goldstein, Classical Mechanics, Addison-Wesley, 2nd. edition, Reading, MA, 1980. |
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O. M. Perdomo,
A dynamical interpretation of cmc Twizzlers surfaces, Pacific J. Math., 258 (2012), 459-485.
doi: 10.2140/pjm.2012.258.459. |
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O. M. Perdomo,
Helicoidal minimal surfaces in $\mathbb{R}^3$, Illinois J. Math., 57 (2013), 87-104.
doi: 10.1215/ijm/1403534487. |
[5] |
O. M. Perdomo,
Constant-speed ramps, Pacific J. Math., 275 (2015), 1-18.
doi: 10.2140/pjm.2015.275.1. |
[6] |
A. P. Usher, A History of Mechanical Inventions, Revised Edition. Dover, 1989. |
[7] |
show all references
References:
[1] |
J. Bertrand,
Théorème relatif au mouvement d'un point attiré vers un centre fixe, C. R. Acad. Sci., 77 (1873), 849-853.
|
[2] |
H. Goldstein, Classical Mechanics, Addison-Wesley, 2nd. edition, Reading, MA, 1980. |
[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. |
[4] |
O. M. Perdomo,
Helicoidal minimal surfaces in $\mathbb{R}^3$, Illinois J. Math., 57 (2013), 87-104.
doi: 10.1215/ijm/1403534487. |
[5] |
O. M. Perdomo,
Constant-speed ramps, Pacific J. Math., 275 (2015), 1-18.
doi: 10.2140/pjm.2015.275.1. |
[6] |
A. P. Usher, A History of Mechanical Inventions, Revised Edition. Dover, 1989. |
[7] |








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