A Study of Super Nonlinear Motion of Electrostatically Coupled Two-Particle System
Author(s)
Haiduke Sarafian
Abstract
We consider a pair of nonidentical mechanical pendulums. The bob of each pendulum in addition to its own mass electrically is charged. The pendulums are hung from a common pivot in a vertical plane forming a slanted asymmet-ric ??shaped figure. For arbitrary initial swings that are not necessarily confined to small angles, we analyze the dy-namics of each bob under the influence of gravity’s pull as well as the mutual repulsive Coulombian internal force. The equations describing the motion of the system are a set of highly, supper nonlinear coupled differential equations. Applying Mathematica we solve the equations numerically. For nonidentical parameters describing the pendulums, namely, we show the system behaves chaotically; i.e. the angular position of each pendulum leaves a non-repeatable, chaotic pattern in time. For this coupled two-particle interactive system we show also by folding the time axis, the angular position of one of the pendulums vs. the other traces a Lissajous type curve. Our report includes various traditional phase diagrams and a set of newly designed, useful, phase-type diagrams as well. For a comprehensive understanding about the dynamics of the problem at hand, we provide Mathematica codes conducive to animating the chaotic motion of the system. The generic format of the codes allows adjusting the relevant pa-rameters at will and addressing the “what-if” scenarios.
We consider a pair of nonidentical mechanical pendulums. The bob of each pendulum in addition to its own mass electrically is charged. The pendulums are hung from a common pivot in a vertical plane forming a slanted asymmet-ric ??shaped figure. For arbitrary initial swings that are not necessarily confined to small angles, we analyze the dy-namics of each bob under the influence of gravity’s pull as well as the mutual repulsive Coulombian internal force. The equations describing the motion of the system are a set of highly, supper nonlinear coupled differential equations. Applying Mathematica we solve the equations numerically. For nonidentical parameters describing the pendulums, namely, we show the system behaves chaotically; i.e. the angular position of each pendulum leaves a non-repeatable, chaotic pattern in time. For this coupled two-particle interactive system we show also by folding the time axis, the angular position of one of the pendulums vs. the other traces a Lissajous type curve. Our report includes various traditional phase diagrams and a set of newly designed, useful, phase-type diagrams as well. For a comprehensive understanding about the dynamics of the problem at hand, we provide Mathematica codes conducive to animating the chaotic motion of the system. The generic format of the codes allows adjusting the relevant pa-rameters at will and addressing the “what-if” scenarios.
Cite this paper
nullH. Sarafian, "A Study of Super Nonlinear Motion of Electrostatically Coupled Two-Particle System," Journal of Electromagnetic Analysis and Applications, Vol. 2 No. 10, 2010, pp. 588-593. doi: 10.4236/jemaa.2010.210076.
nullH. Sarafian, "A Study of Super Nonlinear Motion of Electrostatically Coupled Two-Particle System," Journal of Electromagnetic Analysis and Applications, Vol. 2 No. 10, 2010, pp. 588-593. doi: 10.4236/jemaa.2010.210076.
References
[1] H. Sarafian, “A Study of Super Nonlinear Motion of a Simple Pendulum,” The Mathematica Journal, Vol. 11, in press.
[2] H. Sarafian, “A Study of Super Nonlinear Motion of a Simple Pendulum and Its Generalization,” International Conference on Computational Science and Its Applications, Yongin, 2009, pp. 97-103.
[3] F. Takens, “Detecting Strange Attractors in Turbulence,” In: D. A. Rand and L. S. Young, Eds., Lecture Notes in Mathematics, Springer-Verlag, New York, 1981.