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Constant-speed ramps for a central force field

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

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  • Figure 1.  A mass $ M $ sliding along $ \alpha $ under the effect of a central force $ {\mathbf F}({\mathbf r}) $ and the friction force

    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 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 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 $

  • [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] Wolfram Mathematica 7 Documentation.
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