Figure 1.
A mass $ M $ sliding along $ \alpha $ under the effect of a central force $ {\mathbf F}({\mathbf r}) $ and the friction force
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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 |
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.
Mathematics Subject Classification:
70E18, 53A17.
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
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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. |
| [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] |
Wolfram Mathematica 7 Documentation. Google Scholar |
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. Google Scholar |
| [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] |
Wolfram Mathematica 7 Documentation. Google Scholar |
Figure 2.
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
Figure 3.
Case $ v = 1 $ . The purple part of the logarithmic spiral (left) is the TreadmillSled of the non-circular ramp (right)
Figure 4.
The phase portrait of the system (21). Left: $ v = 2 $ and $ \mu = 0.1 $ . Right: $ v = 0.5 $ and $ \mu = 0.3 $
Figure 5.
TreadmillSleds that are half-lines. Left: $ v = 2 $ , $ \mu = 0.1 $ . Right: $ v = 0.5 $ , $ \mu = 0.3 $
Figure 5, 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} $
Figure 5, 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} $
Figure 8.
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
Figure 9.
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
Figure 10.
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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