Figure 1.
A mass $ M $ sliding along $ \alpha $ under the effect of a central force $ {\mathbf F}({\mathbf r}) $ and the friction force
-
Previous Article
Ergodic pairs for degenerate pseudo Pucci's fully nonlinear operators
- DCDS Home
- This Issue
-
Next Article
A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains
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
![]()
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 $
| [1] |
Alessandro Fonda, Antonio J. Ureña. Periodic, subharmonic, and quasi-periodic oscillations under the action of a central force. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 169-192. doi: 10.3934/dcds.2011.29.169 |
| [2] |
Huaiyu Jian, Hongjie Ju, Wei Sun. Traveling fronts of curve flow with external force field. Communications on Pure & Applied Analysis, 2010, 9 (4) : 975-986. doi: 10.3934/cpaa.2010.9.975 |
| [3] |
Renjun Duan, Shuangqian Liu. Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force. Kinetic & Related Models, 2013, 6 (4) : 687-700. doi: 10.3934/krm.2013.6.687 |
| [4] |
José A. Carrillo, Young-Pil Choi, Yingping Peng. Large friction-high force fields limit for the nonlinear Vlasov–Poisson–Fokker–Planck system. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021052 |
| [5] |
Xianchao Wang, Jiaqi Zhu, Minghui Song, Wei Wu. Fourier method for reconstructing elastic body force from the coupled-wave field. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021052 |
| [6] |
Raffaele Esposito, Yan Guo, Rossana Marra. Validity of the Boltzmann equation with an external force. Kinetic & Related Models, 2011, 4 (2) : 499-515. doi: 10.3934/krm.2011.4.499 |
| [7] |
Joshua E.S. Socolar. Discrete models of force chain networks. Discrete & Continuous Dynamical Systems - B, 2003, 3 (4) : 601-618. doi: 10.3934/dcdsb.2003.3.601 |
| [8] |
Jürgen Scheurle, Stephan Schmitz. A criterion for asymptotic straightness of force fields. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 777-792. doi: 10.3934/dcdsb.2010.14.777 |
| [9] |
Šárka Nečasová. Stokes and Oseen flow with Coriolis force in the exterior domain. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 339-351. doi: 10.3934/dcdss.2008.1.339 |
| [10] |
Mohcine Chraibi, Ulrich Kemloh, Andreas Schadschneider, Armin Seyfried. Force-based models of pedestrian dynamics. Networks & Heterogeneous Media, 2011, 6 (3) : 425-442. doi: 10.3934/nhm.2011.6.425 |
| [11] |
Fei Meng, Xiao-Ping Yang. Elastic limit and vanishing external force for granular systems. Kinetic & Related Models, 2019, 12 (1) : 159-176. doi: 10.3934/krm.2019007 |
| [12] |
Hongjun Yu. Global classical solutions to the Boltzmann equation with external force. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1647-1668. doi: 10.3934/cpaa.2009.8.1647 |
| [13] |
Yingzhe Fan, Yuanjie Lei. The Boltzmann equation with frictional force for very soft potentials in the whole space. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 4303-4329. doi: 10.3934/dcds.2019174 |
| [14] |
Taige Wang, Bing-Yu Zhang. Forced oscillation of viscous Burgers' equation with a time-periodic force. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1205-1221. doi: 10.3934/dcdsb.2020160 |
| [15] |
Joyce R. McLaughlin and Arturo Portnoy. Perturbation expansions for eigenvalues and eigenvectors for a rectangular membrane subject to a restorative force. Electronic Research Announcements, 1997, 3: 72-77. |
| [16] |
Yiqian Wang. Boundedness of solutions in a class of Duffing equations with a bounded restore force. Discrete & Continuous Dynamical Systems, 2006, 14 (4) : 783-800. doi: 10.3934/dcds.2006.14.783 |
| [17] |
Zhigang Wu, Wenjun Wang. Uniform stability of the Boltzmann equation with an external force near vacuum. Communications on Pure & Applied Analysis, 2015, 14 (3) : 811-823. doi: 10.3934/cpaa.2015.14.811 |
| [18] |
Takeshi Taniguchi. The exponential behavior of Navier-Stokes equations with time delay external force. Discrete & Continuous Dynamical Systems, 2005, 12 (5) : 997-1018. doi: 10.3934/dcds.2005.12.997 |
| [19] |
Daniel Han-Kwan. $L^1$ averaging lemma for transport equations with Lipschitz force fields. Kinetic & Related Models, 2010, 3 (4) : 669-683. doi: 10.3934/krm.2010.3.669 |
| [20] |
Mi-Ho Giga, Yoshikazu Giga. A subdifferential interpretation of crystalline motion under nonuniform driving force. Conference Publications, 1998, 1998 (Special) : 276-287. doi: 10.3934/proc.1998.1998.276 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]