Given the equation motion of a moving charged particle in a controlled electromagnetic field, this paper proves that its velocity-trajectory motion converges to an specified velocity-surface in the 3-D Euclidean dimensional space. This is basically realized by just manipulating the electric field of an electromagnetic field. Lyapunov theory is invoked to test our statement; besides, a numerical example is provided to support our theoretical contribution. Finally, we consider that the exposition of this paper could be of interest to undergraduate students.Peer ReviewedPostprint (author's final draft

Acho Zuppa, Leonardo

English

Given the equation motion of a moving charged particle in a controlled electromagnetic field, this paper proves that its velocity-trajectory motion converges to an specified velocity-surface in the 3-D Euclidean dimensional space. This is basically realized by just manipulating the electric field of an electromagnetic field. Lyapunov theory is invoked to test our statement; besides, a numerical example is provided to support our theoretical contribution. Finally, we consider that the exposition of this paper could be of interest to undergraduate students.Peer Reviewe

UPCommons

Velocity-surface stability of a moving charged particle in a controlled electromagnetic field

UPCommons. Portal del coneixement obert de la UPC

Physics Education 1 Apr-Jun 2016Velocity-surface stability of a moving charged particle ina controlled electromagnetic fieldL. Acho11Department of MathematicsPolytechnic University of CatalunyaBarcelona 08036, Spain.leonardo.acho@upc.edu(Submitted 23-12-2015)AbstractGiven the equation motion of a moving charged particle in a controlled electromagneticfield, this paper proves that its velocity-trajectory motion converges to an specifiedvelocity-surface in the 3-D Euclidean dimensional space. This is basically realized by justmanipulating the electric field of an electromagnetic field. Lyapunov theory is invoked totest our statement; besides, a numerical example is provided to support our theoreticalcontribution. Finally, we consider that the exposition of this paper could be of interest toundergraduate students.1 IntroductionThe second Lyapunov method, also calledThe Direct Lyapunov Method, is now widelyused to analyse stability motion of dynamicsystems due to its simplicity and efficiency.This theory has been applied, for instance, inphysics, astronomy, chemistry, biology, andso on. The main benefit of Lyapunov theoryis its ability to conclude stability of systems(stability in the sense of Lyapunov) withoutexplicitly integrating the differential equa-tions involved in our system. This methodis basically a generalization of the idea that aenergy function, the Lyapunov function, asso-ciated to a system to be analysed, is decreas-Volume 32, Number 2, Article Number : 6 www.physedu.inPhysics Education 2 Apr-Jun 2016ing along the system trajectory. For a briefdiscussion on this theory, see, for instance [1].On the other hand, and motivated by ap-plications where the natural operation modeis periodic, for instance, orbital stabilizationof mechanical systems (see, e.g., [2] and refer-ences therein), stability analysis of a velocity-surface for a moving charged particle in aelectromagnetic environment is here analysedby using Lyapunov theory. That is to say,given the motion of a travelling charged par-ticle in a controlled electromagnetic field, weprove that its velocity-trajectory converges toan specified velocity-surface in the 3-D Eu-clidean dimensional space. At last to say, theuse of a Lyapunov function to conclude or-bital stability of dynamical systems is alsocovered in [3].The rest of this paper is organized as fol-lows. Section II gives the problem statementand a solution to it. Section III shows a nu-merical experiment to support our theoreti-cal affirmation. Finally, Section IV states theconclusions.2 Velocity-surfacestability analysisThe objective of this section is to demon-strate that it is possible to produce a 3-D-velocity-surface such that any velocity-trajectory of a charged particle in motionstarting close enough to this surface, will con-verge to it as time goes on. This is realized bymanipulating the electric field of an electro-magnetic field. Lyapunov theory is the math-ematical tool employed to stablish this state-ment.The equation describing the motion of acharged particle Q in a electromagnetic fieldsis defined bymdvdt= Q[v×B+ E], (1)where B and E are the magnetic and electricfields, respectively; v is the velocity of thecharged particle in this electromagnetic envi-ronment, and m is the particle mass. Let ussuppose that the electric field (a controlledone) isE = v(k− | v |), (2)where | · | is the vector Euclidean norm, andk ∈ R is a positive parameter at our disposal.Then, equation (2) in (1) producesmdvdt= Q [v×B+ v(k− | v |)] . (3)Let us use the next candidate Lyapunovfunction:V (t) = v · v. (4)Then, its time-derivative along the systemtrajectory (3) yieldsV˙ (t) =dV (t)dt=2Qm[v · (v×B) + v · v(k− | v |)]=2Qm[v · v(k− | v |)] . (5)Volume 32, Number 2, Article Number : 6 www.physedu.inPhysics Education 3 Apr-Jun 2016Observe that V˙ (t) is positive if | v |< kand negative if | v |> k. This meansthat the velocity-surface equation (VSE),| v |= k, is locally attractive; i.e., for anyvelocity-trajectory starting close enough tothis velocity-surface, the system trajectory(3) will converge to it. On the other hand,if the origin of the system (3) is the onlyequilibrium point; then, any trajectory, butthe origin, will converge to the VSE1. Inresume, we arrive to the following result.Theorem 1.- Given the dynamic of acharged particle in motion (1), its velocity-trajectory will locally converge to thevelocity-surface established by | v |= k,k ∈ R+, if we set the electric field as E =v(k− | v |) in an electromagnetic field. More-over, if the system only has the origin as itsunique equilibrium point; then, any velocity-trajectory starting anywhere but the originwill converge to the stated velocity-surface.3 Numerical exampleThis section describes a numerical example.Hence, if v = vxi+ vyj+ vzk, and B = Bxi+Byj + Bzk, then the corresponding ordinarydifferential equations of (3), yieldsv˙x =Qm(Bzvy − vzBy + pvx),v˙y =Qm(Bxvz − vxBz + pvy),1Almost the same lines of thinking are employedby [3] to conclude orbital stability in one of the givenexamples.v˙z =Qm(Byvx − vyBx + pvz),where p = k −√v2x + v2y + v2z . Using k = 2and the charged particle with Q = 8µc andmass m = 0.5µ kg, simulation results of theabove system are shown in Figures 1, 2 ad 3.Each figure present the same experiments us-ing trajectories generated by three differentsets of initial conditions. The blue line cor-responds to vx(0) = −3.5, vy(0) = 0.0, andvz(0) = 3.5; the red line for the case whenvx(0) = 0.0, vy(0) = 0.0, and vz(0) = −1.5;and the yellow one with vx(0) = 0.0, vy(0) =−4.0, and vz(0) = 0.0. All of them in meterper second. Finally, a free space motion isassumed. From these figures, we can appre-ciate that these trajectories converge to theexpected velocity-spherical-surface (the greensurface shown in the mentioned figures).−4−2024−4−2024−4−2024vv vxyzFigure 1: Simulation result using Bx = Bz =0.02T and By = −0.02T.Volume 32, Number 2, Article Number : 6 www.physedu.inPhysics Education 4 Apr-Jun 2016−4−2024−4−2024−4−2024vv vxyzFigure 2: Simulation result using By = Bz =0.02T and Bx = −0.02T.−4−2024−4−2024−4−2024vv vxyzFigure 3: Simulation result using Bx = Bz =−0.02T and By = 0.02cos(0.002t)T.4 ConclusionLyapunov theory was used to prove stabil-ity of a specified velocity-surface of a movingcharged particle in a controlled electromag-netic field. We assumed that the electric fieldis manipulable, and we set E = v(k− | v |).Obviously, and according to the Lyapunovtheory, other velocity surfaces are possible bychanging the term | v |, for instance, to an el-lipsoid, if | v | is replaced by√v2xa2+v2yb2+ v2zc2,and so far.AcknowledgmentsThis work is partially supported by the Span-ish Ministry of Economy and Competitive-ness through grant DPI2015-64170-R.References[1] S. Siboni (2008). Am J. Phys. 76, 649[2] Y. Orlov, L. Aguilar, L. Acho (2008).Int. J. of Cntrol 81(2), 227[3] R. L. Borelli, C. S. Coleman Differen-tial equations: a modelling perspective,(John Wiley and Sons, Inc., New York,1998) Chap. 9Volume 32, Number 2, Article Number : 6 www.physedu.in

https://upcommons.upc.edu/bitstream/2117/91267/1/6.-Velocity-surface-stability-of-a-moving-charged-particle--in-a.pdf

Velocity-surface stability of a moving charged particle in a controlled electromagnetic field

Abstract

Similar works

Full text

Available Versions

UPCommons

UPCommons. Portal del coneixement obert de la UPC