Using Mathematica software to solve ordinary differential equations and applying it tothe graphical representation of trajectories

Deyvid W. da M. Pastana(a) and Manuel E. Rodrigues(a,b)

(a)Faculdade de Ciencias Exatas e Tecnologia, Universidade Federal do ParaCampus Universitario de Abaetetuba, 68440-000, Abaetetuba, Para, Brazil

(b)Faculdade de Fısica, Programa de Pos-Graduacao em Fısica,Universidade Federal do Para, 66075-110, Belem, Para, Brazil

(Dated: 9 de abril de 2021)

We discuss the great importance of using mathematical software in solving problems in today’ssociety. In particular, we show how to use Mathematica software to solve ordinary differentialequations exactly and numerically. We also show how to represent these solutions graphically. Wetreat the particular case of a charged particle subject to an oscillating electric field in the xy planeand a constant magnetic field. We show how to construct the equations of motion, defined by thevectors position, velocity, electric and magnetic fields. We show how to solve these equations ofLorentz force, and graphically represent the possible trajectories. We end by showing how to builda video simulation for an oscillating electric field trajectory particle case.

I. INTRODUCTION

In a Bachelor’s or Undergraduate course in Physics thestudent always comes across the subject of ordinary diffe-rential equations early on. This comes from the fact thatNewton’s second law is a second order ordinary differen-tial equation. This law can be integrated exactly, in somecommon cases [11–13], or it may not have an exact analy-tic solution, so it must be integrated numerically [18]. Awell-known special case in which Newton’s second law hasno analytical solution is that of the simple pendulum [18].In this case, the differential equation is ordinary secondorder, but it is also nonlinear, which makes it impossibleto integrate. A powerful method for solving ordinary dif-ferential equations that have no analytic solution is thenumerical integration method. This method solves thegiven equation and provides a valid solution for a giveninitial condition and range of the independent variable.We can then obtain other functions, or quantities suchas velocity and acceleration, from this numerical solution.We can also interpret the given system physically.

In physics and mathematics there has always been aneed to formulate new methods for solving new problems.But there was always a limitation in the question of thepossibility of the number of calculations. This was no-ticed more recently with the attempt to solve the three-body problem. The invention of the computer to facili-tate these calculations was a milestone in physics in sol-ving non-analytical problems. Through this tool, one cansolve and interpret problems that were precluded fromdeeper analysis. The most recent example of the incre-dible amount of calculations done to obtain a result isthe case of the sum of the cubes of three numbers [2, 19].In this case, the problem is formulated as follows: Arethere three real numbers raised to the cube that whenadded together result in the number 42? This may seemlike an easy problem, but it is a long way from being sol-ved manually. Only with the help of a supercomputer,where you can make or check an absurd amount of calcu-

lations using about four hundred thousand home compu-ters, can the problem be solved. The answer found was42 = (−80538738812075974)3 + (80435758145817515)3 +(12602123297335631)3.

The use of the computer [9, 14], with its software, tohelp students in elementary school [15, 16] and highereducation [1, 10], is growing every year, especially in thearea of exact sciences. This tool is already proven effec-tive in solving problems and facilitating student learningin various topics in Physics [3]. Computer simulation alsoaids student learning [5, 7, 17].

Our main goal for this article is to introduce basic co-des in Mathematica software for solving ordinary diffe-rential equations exactly or numerically, and graphicallyrepresenting some particle trajectories that would be im-possible to do manually. It is our intention that afterreading this article carefully, the physics student will beable to solve analytical and numerical problems, as wellas graph their results and interpret them.

The software is widely used to solve differential equati-ons that govern the motion of charged particles subjectedto electric and magnetic fields and to generate simulati-ons of the motion for analysis, as shown in the paper [4],where the authors found different confinement trajecto-ries for the uranium isotope subjected to an oscillatingmagnetic field. So, software is a useful tool for the inves-tigation of magnetic traps.

The paper is presented as follows: in the section IIwe give an introduction to the essential commands ofMathematica software, for analytic and numerical solvingof ordinary differential equations. This is done throughexamples such as the simple harmonic oscillator, simplependulum, and motion with velocity proportional resis-tance, and also shows how to create a simulation movie.In the III section we make full use of Mathematica soft-ware, for the problem of oscillating electric and magneticfields, showing some trajectories that are impossible totrace manually, and the construction of a movie to simu-late these trajectories. In the IV section we make our

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II. MATHEMATICA AS A TOOL FOR SOLVINGDIFFERENTIAL EQUATIONS

In this section we will learn how to use Mathematicasoftware to solve ordinary differential equations exactlyand numerically. We will also show its usefulness ingraphing. We will use the 12.1 version of Mathematicasoftware, but the commands and code can be used in anyprevious or newer version.

One of the first and most important equations in phy-sics is that of the simple harmonic oscillator

x(t) + ω20x(t) = 0 . (1)

This equation has a unique general solution for a gi-ven initial condition, for example x(0) = x0 e x(0) = v0.We can integrate this equation to find an exact solutionby Mathematica software, using the following input com-mand

DSolve[D[D[x[t],t],t]+ω0^2 x[t]==0,x[t],t]

The student should type these commands into a Mathe-matica notebook, hold down Shift and press Enter. Theoutput is given by

{{x[t]->c1Cos[tω0]+c2Sin[tω0]}}

We can add the initial conditions as follows

DSolve [{D[D[x[t],t],t]+ω0^2 x[t]==0,x[0]==0 ,x′

[0]==1} ,x[t],t]

For us to be able to use this analytical solution, wemust match it to some name, so we rewrite

exactsolution=DSolve [{D[D[xe[t],t],t]+ω0^2 xe[t

]==0,xe[0]==0 ,xe′[0]==1} ,xe[t],t]

where xe[t] is the exact solution of the equation. We cangraph this exact solution with the command

Plot[{xe[t]/. exactsolution /.ω0->1},{t,0,10},PlotStyle -> Blue , AxesLabel -> {”t”, ”x(t)”}]

The output image will be that of the figure 1. To call theexact solution we use the command /.solucaoexata, andto assign a numerical value to ω0, we use the command/.ω0− > 1. This same procedure can be used for anyordinary differential equation that has an exact analyticsolution. The student can exercise through a questionfrom a mathematical methods book, such as ProfessorBassalo’s, Arfken’s, or Butkov’s.

2 4 6 8 10t

-1.0

-0.5

0.5

1.0

x(t

Figura 1: Graph of the exact solution x(t).

Now we can solve this same differential equation (1),using a numerical method. To do so, we use the followingcommand

m=1;k=1;

numericalsolution=NDSolve [{m*D[D[x[t],t],t]+k*x[

t]==0,x[0]==0 ,x′[0]==1} ,{x[t]},{t ,0 ,100000}]

We can represent graphically with the command

Plot[Evaluate[x[t] /. numericalsolution], {t, 0,

10}, PlotStyle -> Blue ,AxesLabel -> {”t”, ”x(t)”}]

The output is the same image as the figure 1, becausethe numerical solution has a small difference from theexact one. This can be verified as follows. Let’s define anew error function, which is the subtraction of the exactsolution from the numerical solution

error[t_] :=

Simplify[Evaluate[xe[t] /. exactsolution /. ω0-> 1] - Evaluate[x[t] /. numericalsolution ]]

Agora podemos plotar com

Plot[{error[t]}, {t, 0, 100000} , PlotStyle ->

Blue , AxesLabel -> {”t”, ”error(t)”}]

The output is represented in figure 2. We can see thatthe error function is oscillatory increasing. Up to timet = 105, the error function is bounded in the interval(0.15 × 10−4). It can then be seen that the longer thetime interval, the numerical solution moves further awayfrom the exact one, and the error function grows. Thiscan be mitigated by improving the numerical method ofsolving the differential equation.

3

20000 40000 60000 80000 100000t

-0.0015

-0.0010

-0.0005

0.0005

0.0010

0.0015

erro(t)

Figura 2: Error function graph.

We will now improve the numerical method, using thecommand

numericalsolution2 = NDSolve [{m*x′′[t] + k*x[t]

== 0, x[0] == 0, x′[0] == 1}, {x[t]}, {t

,0 ,100000} , MaxSteps -> Infinity ,

AccuracyGoal -> 10, PrecisionGoal -> 10]

If we define a new error function, with the numerical so-lution 2, we can see that it is restricted to a new interval(0; 4× 10−5), thus improving the accuracy of the nume-rical solution. We can further improve this result. Wedefine a new numerical solution with the command

numericalsolution3 = NDSolve [{m*x′′[t] + k*x[t]

== 0, x[0] == 0, x′[0] == 1}, {x[t]}, {t

,0 ,100000} , MaxSteps -> Infinity ,

AccuracyGoal -> 20, PrecisionGoal -> 20,

WorkingPrecision -> 30]

We should point out here that the computational cost,that is, the time spent by the computer, to obtain the nu-merical solution with this accuracy, is incredibly higher.Thus, defining the error function and its plot by

error3[t_]:= Simplify[Evaluate[xe[t] /.

exactsolution /. ω0 -> 1] - Evaluate[x[t] /.

numericalsolution3 ]]

Plot[{erro3[t]}, {t, 0, 100000} , PlotStyle ->

Blue , AxesLabel -> {”t”, ”erro(t)”}]

The output is represented in figure 3. We can see thatthe error3 function is oscillatory. Up to time t = 105, theerror3 function is bounded in the interval (0; 8× 10−12).This shows that this numerical method is very reliable,with an error of the order of 10−12.

20000 40000 60000 80000 100000t

-6.×10-12

-4.×10-12

-2.×10-12

2.×10-12

4.×10-12

6.×10-12

8.×10-12

Figura 3: Graph of the error3 function.

Now we can calculate the velocity from the exact so-lution

velocity[t_] := Evaluate[D[xe[t] /.

exactsolution , t]]

We can plot with the command

Plot[{ velocity[t] /. \[Omega]0 -> 1}, {t, 0,

50}, PlotStyle -> Blue , AxesLabel -> {”t”, ”v(t)”}]

We could use the numerical3 solution, like this

velocityN[t_] := Evaluate[D[x[t] /.

numericalsolution , t]]

By defining an error function for the velocity, we can seethat the derivatives, or velocities, differ by the same orderof magnitude as the original functions xe(t) and x(t).

One important tool is to make a parametric plot.We know that the position and velocity of a simpleharmonic oscillator depend explicitly on time. So wecan make a parametric plot x′ × x. For example, ifxe(t) = A cos(ω0t), ve(t) = −Aω0 sin(ω0t), then by thefundamental theorem of trigonometry sin2 t+ cos2 t = 1,we have [ve/(Aω0)]2 +[xe/A]2 = 1, which is the equationof a circle. So doing the commands

ParametricPlot [{xe[t], velocity[t]} /.

exactsolution /. ω0 -> 1, {t, 0, 50},

AxesOrigin -> {0, 0}, AxesLabel -> {x[t], x′

[t]}, PlotRange -> All , ImageSize -> 300,

PlotStyle -> Blue]

we have the figure 4.

4

-1.0 -0.5 0.5 1.0x(t)

-1.0

-0.5

0.5

1.0

x′

Figura 4: Parametric plot of x′ × x.

Now as a practical and simple example, we can presentthe equation of a simple pendulum

θ(t) +g

Lsin [θ(t)] = 0 , (2)

where g is the acceleration of gravity, L is the length ofthe pendulum’s string and θ(t) is the angle the stringmakes with the vertical. As said before, the equation (2)is a nonlinear second order ordinary differential equation,which makes it impossible to integrate analytically. Wecan then use the numerical integration method presentedabove. Using the commands

g = 9.81; L = 0.2;

numericalsolution = NDSolve [{θ′′[t] + (g/L)*Sin[θ[t]] == 0, θ[0] == π/4, θ′[0] == 0}, {θ[t]},{t ,0 ,10000}]

we get the numerical solution for the simple pendulum.We can calculate the velocity of the pendulum by writing

v[t_] := Evaluate[D[θ[t] /. numericalsolution , t

]]

then we represent the parametric graph by

ParametricPlot [{θ[t], v[t]} /. numericalsolution

, {t, 0, 1}, AxesOrigin -> {0, 0}, AxesLabel

-> {”θ(t)”, ”θ′(t)”}, PlotRange -> All ,

ImageSize -> 100, PlotStyle -> Blue]

resulting in the figure 5.

-0.5 0.5�(t)

-4

-2

2

4

�'

Figura 5: Parametric plot of θ′ × θ.

Finally, we will create a movie with the trajectorygraphs of a particle. We will take uniform circular mo-tion, so x(t) = A cos(ωt) and y(t) = A sin(ωt). ChoosingA = ω = 1, we can create a movie with the followingcommands

x[t_] := A*Cos[ω*t]; y[t_] := A*Sin[ω*t]; A = 1;

ω = 1; tRange = Range [0.001 , 20*Pi, 20*Pi

/500];

For[k = 1, k <= Length[tRange], k++, tk = tRange

[[k]]; xk = x[tk]; yk = y[tk];

movieVector[k] =

Show[Graphics [{ AbsolutePointSize [15], Green ,

Point[{xk, yk}]},

PlotRange -> {{-1.1, 1.1}, {-1.1, 1.1}}] ,

ParametricPlot [{x[t], y[t]}, {t, 0, tk},

PlotStyle -> Red ,

PlotRange -> {{-1.1, 1.1}, {-1.1, 1.1}}] ,

AxesLabel -> {”x”, ”y”},PlotLabel -> StringJoin[”t=”, ToString[N[tk

]]]]]

M = Table[movieVector[k], {k, 1, 500}];

fileName = ”FilmeMathematica.avi”; Export[

fileName , M]

Directory []

5

The last command, Directory[], will show you wherethe file is located on your computer. The gene-rated file is in avi format. We submit the re-sult on Youtube at the following electronic address:https://youtu.be/MLv10LFUhjA.

Now we can move on to the next section, where wewill show you some more Mathematica software toolsand some trajectories where it would be impossible torepresent them manually.

III. MATHEMATICA APPLIED TO CLASSICALELECTROMAGNETISM

Electromagnetism is a subject that underlies the the-ory of electric and magnetic fields, which are generatedby sources and currents as stated in Maxwell’s equations,

∇ · ~E =ρ

ε0(3a)

∇ · ~B = 0 (3b)

∇× ~E = −∂~B

∂t(3c)

∇× ~B = µ0~J + µ0ε0

∂ ~E

∂t(3d)

Once a system is established, the conservation of char-ges locally is represented by the continuity equation,

∇ · ~J +∂ρ

∂t= 0. (4)

Therefore, Maxwell’s equations govern the form fieldscan take such that if an electric field is reproduced expe-rimentally, it will in turn produce a magnetic field cor-responding with Maxwell’s equations, and the same canbe said for current density and charge desity by meansof the continuity equation.

Given a magnetic field and an electric field, we cancheck whether they satisfy Maxwell’s equations and thecontinuity equation easily using Mathematica softwareby using the codes

Div[E,{x, y, z}] == ρ/ε0

Div[B,{x, y, z}] == 0

Curl[E,{x, y, z}] == -D[B,t]

Curl[B,{x ,y ,z }] == µ0J+µ0ε0D[E,t]

Div[J,{x, y, z}]+D[ρ,t] == 0

where the commands Div[] e Curl[] represent the diver-gent and rotational, respectively, of the fields.

The effects of electric and magnetic fields on an electriccharge Q are represented by the Lorentz force,

~F = Q(~E + ~v × ~B

). (5)

Equating the (??) to Newton’s second law of motion,we obtain the equations of motion for this system, withwhich we can analyze the dynamics of a charged particleunder the action of electric and magnetic fields.

A. Oscillating fields and charged particles

In this section we will show how Mathematica softwarecan be used to solve the equations of motion and gene-rate simulations of possible trajectories for an electricallycharged particle subjected to an oscillating electric fieldand a uniform magnetic field.

1. ~E oscillating (x) and ~B uniform (z)

Consider a charged particle of mass M and charge Q,subjected to the fields

~E = E0 cos(ωt)i (6)

~B =Mω0

Qk (7)

Note that the fields (6) and (7) do not satisfy theAmpere-Maxwell law, equation (3d), in the absence ofsources and currents. However, all of Maxwell’s equati-ons are satisfied if one validates

~J = ε0E0ω sin(ωt)i. (8)

This is in accordance with the continuity equation (4),because the current density vector (8) is uniform andtherefore has zero divergence.

An electric field oscillating in time and uniform inspace satisfies all Maxwell’s equations for any constant

magnetic field if ~J = −∂ ~E/∂t. This condition immedia-

tely implies, ∇ · ~J = 0. Given that

i =

∫V

∇ · ~JdV, (9)

no current located within a volume V exits by a flowthrough a surface A, or, furthermore, an external cur-rent does not enter within the volume of considerationV through a surface A. Thus, there is no variation ofa net charge density as a function of spatial variables,which is characteristic of a stationary magnetic field [8].Therefore, the current density (8) obeys the laws of mag-netostatics, as it must, since we are considering a uniformmagnetic field.

Replacing the fields (6) and (7) in the Lorentz force,we obtain the equations of motion

QE0 cos(ωt) +M(ω0y − x) = 0 (10a)

−M (ω0x+ y) = 0 (10b)

−Mz = 0. (10c)

6

Given the initial conditions x(0) = y(0) = z(0) =x(0) = y(0) = z(0) = 0, we can solve the coupled ODEsystem (10), obtaining the solutions

x(t) =E0Q[cos(ω0t)− cos(ωt)]

M(ω2 − ω20)

(11)

y(t) =E0Q [ω0 sin(ωt)− ωsin(ω0t)]

M(ω3 − ωω20)

(12)

z(t) = 0. (13)

Setting the parameters M = E0 = ω0 = Q = 1 we canexpress the motions of the charged particle, which occurin the xy plane, for different values of homega, whensubjected to the fields (6) and (7), as shown in the figure6.

ω=0.1 ω=0.433333 ω=0.766667

ω=1.1 ω=1.43333 ω=1.76667

ω=2.1 ω=2.43333 ω=2.76667

ω=3.1 ω=3.43333 ω=3.76667

Figura 6: Trajectories of a charged particle subjectedto an oscillating electric field and a uniform magneticfield, with the oarameter M = E0 = ω0 = Q = 1and initial conditions x(0) = y(0) = z(0) = x(0) =y(0) = z(0) = 0.

Using Mathematica software, this procedure can be ve-rified, along with obtaining graphs, from the commands

F = {Fx,Fy ,Fz};r = {x[t],y[t],z[t]};EF = {Ex

,Ey ,Ez};BF = {Bx,By ,Bz};v = D[r,t];

F = Q*(EF+Cross[v,BF])

eq1 = F-M*D[r,{t,2}] == 0// Thread; eq1//

ColumnForm

initial = {x[0] == 0,x’[0] == 0,y[0] == 0,y

’[0] == 0,z[0] == 0,z’[0] == 0};

fields = Thread/@{BF ->{0,0,M*ω0/Q},EF ->{E0*\

cos[ω*t],0,0}}// Flatten

eqs = Join[eq1 ,inicial] /. fields

solution = DSolve[eqs ,{x[t],y[t],z[t]},t]//

Flatten // FullSimplify

values = {M->1,E0 ->1,ω0->1,Q->1}

graphic[t_,ω] := {x[t],y[t],z[t]}/. solution

/. values

graphic[ω_] := ParametricPlot[point[t,ω][[{1 ,2}]]// Evaluate ,{t,0 ,20*π},PlotLabel ->

StringJoin ["ω=", ToString[N[ω]]],Ticks ->None ,DisplayFunction ->Identity , PlotStyle ->

Blue]

table := Table[graphic[ω],{ω ,0.1,4,1/3}]

Show[GraphicsArray[Partition[table ,3]]]

where Table[grafico[ω],ω,0.1,4, 1/3] is a command thatcollects the points generated by the solutions, for dif-ferent values ω, building a table, which is called withShow[GraphicsArray[Partition[tabela,3]]] to generate thegraphs of the trajectories illustrated in the figure 6.

2. ~E oscillating in two directions (xy) and ~B uniform in a(z)

Given a particle of charge Q and mass M under theaction of the fields

~E = E0 cos(ωt)i+ E0 sin(ωt)j (14)

~B =Mω0

Qk, (15)

The equations of motion for this system will be

QE0 cos(ωt) +M(ω0y − x) = 0 (16a)

QE0 sin(ωt)−M (ω0x+ y) = 0 (16b)

−Mz = 0. (16c)

The System (16), considering the initial conditionsx(0) = y(0) = z(0) = x(0) = y(0) = z(0) = 0, has asa general solution

x(t) =E0Q [ω + ω0 − ω0 cos(tω)− ω cos(tω0)]

(mω0ω0(ω0 + ω0))(17)

y(t) =E0Q [ω sin(tω0)− ω0 sin(tω)]

mωω0(ω + ω0)(18)

z(t) = 0 (19)

7

By defining the parameters M = E0 = ω0 = Q = 1,we obtain the trajectories that the particle describes in aplane, for different values of ω, in the xy plane, as shownin the figure 7.

ω=0.1 ω=0.433333 ω=0.766667

ω=1.1 ω=1.43333 ω=1.76667

ω=2.1 ω=2.43333 ω=2.76667

ω=3.1 ω=3.43333 ω=3.76667

Figura 7: Trajectories of a charged particle subjectedto an oscillating electric field in two directions and auniform magnetic field in one, with the parametersM = E0 = ω0 = Q = 1 and initial conditions x(0) =y(0) = z(0) = x(0) = y(0) = z(0) = 0.

The trajectories illustrated in the figure 7 were gene-rated in Mathematica from the commands

F = {Fx, Fy, Fz}; r = {x[t], y[t], z[t]}; EF

= {Ex, Ey ,Ez}; BF = {Bx, By , Bz}; v = D[r,

t];

F = Q*(EF + Cross[v, BF])

eq1 = F - M*D[r, {t, 2}] == 0 // Thread; eq1

// ColumnForm

initial = {x[0] == 0, x’[0] == 0, y[0] == 0,

y’[0] == 0, z[0] == 0, z’[0] == 0};

fields = Thread /@ {BF ->{0, 0, m*ω0/q},EF ->{

E0*Cos[ω*t], E0*Sin[ω*t], 0}} // Flatten

eqs = Join[eq1 , initial] /. fields

solution = DSolve[eqs , {x[t], y[t], z[t]}, t

] // Flatten // FullSimplify

values = {M -> 1, E0 -> 1, ω0 -> 1, Q -> 1}

point[t_, ω_] := {x[t], y[t], z[t]} /.

solution /. values

graphic[ω_]:= ParametricPlot[point[t,ω][[{1 ,2}]]// Evaluate ,{t,0 ,20*Pi},PlotLabel ->

StringJoin ["ω=",ToString[N[ω]]],Ticks ->None ,DisplayFunction ->Identity , PlotStyle -> Blue

]

table := Table[graphic[ω], {ω, 0.1, 4, 1/3}]

Show[GraphicsArray[Partition[tabela , 3]]]

3. ~E oscillating in two directions (xy) and ~B uniform (x)

Let’s consider the electric field used in the case III A 2,this time with a constant magnetic field Bx in the direc-tion of the x axis, i.e.,

~E = E0 cos(ωt)i+ E0 sin(ωt)j (20)

~B = Bxi (21)

A system in which a charged particle of mass M andcharge Q, subjected to the action of the fields (20) and(21), leads to the equations of motion

E0Q cos(ωt)−Mx = 0 (22a)

Q [E0 sin(ωt) +Bxz]−My = 0 (22b)

−BxQy −Mz = 0 (22c)

Assuming that the initial conditions are x(0) = y(0) =z(0) = x(0) = y(0) = z(0) = 0, we obtain the followingsolution

x(t) = −E0Q cos(ωt)− 1

Mω2(23)

y(t) =E0M [−Mω sin(BxQt/M) +BxQ sin(ωt)]

B3xQ

2 −BxM2ω2(24)

z(t) =E0(−B2

xQ2 +M2ω2 −M2ω2 cos(BxQt/M)

B3xQ

2ω −BxM2ω3

+E0(B2

xQ2 cos(ωt))

B3xQ

2ω −BxM2ω3(25)

Defining the values of the constants as E0 = Q = M =Bx = 1, we get the trajectories shown in the figures 8, 9and 10.

8

Figura 8: Trajectory of a charged particle subjectedto fields (20) and (21), with the parameters M =E0 = Bx = Q = 1, ω = 1, 5 and initial conditionsx(0) = y(0) = z(0) = x(0) = y(0) = z(0) = 0, in atime interval [0, 20π].

Figura 9: Trajectory of a charged particle subjectedto fields (20) and (21), with the parameters M =E0 = Bx = Q = 1, ω = 2.1 and initial conditionsx(0) = y(0) = z(0) = x(0) = y(0) = z(0) = 0, in atime interval [0, 20π].

Figura 10: Trajectory of a charged particle subjectedto fields (20) and (21), with the parameters M =E0 = Bx = Q = 1, ω = 0.51 and initial conditionsx(0) = y(0) = z(0) = x(0) = y(0) = z(0) = 0, in atime interval [0, 100π].

The solutions (23), (24) and (25), can be obtained inMathematica from the commands

F = {Fx, Fy, Fz}; r = {x[t], y[t], z[t]}; EF

= {Ex, Ey , Ez}; BF = {Bx, By, Bz}; v = D[r,

t];

F = Q*(EF + Cross[v, BF])

eq1 = F - M*D[r, {t, 2}] == 0 // Thread; eq1

// ColumnForm

initial = {x[0] == 0, x’[0] == 0, y[0] == 0,

y’[0] == 0, z[0] == 0, z’[0] == 0};

fields = Thread /@ {BF -> {Bx, 0, 0}, EF ->

{E0*Cos[ω*t],E0*Sin[ω*t], 0}} // Flatten

eqs = Join[eq1 , initial] /. fields

DSolve[eqs , {x[t], y[t], z[t]}, t] //

Flatten // FullSimplify

Once the solutions are obtained, we use Parametric-Plot3D to build the computer simulation of the motionof the charged particle illustrated in figure 8, from thecodes

values = {M -> 1, E0 -> 1, Bx -> 1, Q -> 1}

eqx[t_] := -((E0 q (-1+Cos[t ω]))/(m ω^2));eqy[t_] := (E0 m (-m ω Sin[(Bx q t)/m] + Bx

q Sin[t ω]))/(Bx^3 q^2 - Bx m^2 ω^2);eqz[t_] := (E0 (-Bx^2 q^2 + m^2 ω^2 - m^2 ω

9

^2 Cos[(Bx q t)/m] + Bx^2 q^2 Cos[t ω]))/(Bx^3 q^2 ω - Bx m^2 ω^3);

ParametricPlot3D [{eqx[t], eqy[t], eqz[t]} /.

ω -> 1.5 /. valores , {t, 0, 20*Pi},

AxesLabel -> {"x", "y", "z"}, Mesh -> None ,

PlotRange -> All , PlotStyle -> {Red ,

Directive[EdgeForm []], Thickness [0.005] ,

Opacity [1]}, ImageSize -> Medium]

The figures 9 and 10, are easily obtained by varyingthe value of ω set in the ParametricPlot3D command asω −> 1.5, for the values 2.1, e 0.51, respectively.

4. ~E oscillating in two directions (xy) and ~B uniform in a(y)

Consider a particle of mass M and charge Q subjectedto the effects of an oscillating electric and magnetic fielduniform in the direction of the axis x,

~E = E0 cos(ωt)i+ E0 sin(ωt)j (26)

~B = By j. (27)

The equations of motion for the system formed by theparticle and the fields (26) and (27) are

Q [E0 cos(ωt)−By z]−Mx = 0 (28a)

QE0 sin(ωt)−My = 0 (28b)

QByx−Mz = 0 (28c)

Assuming that the initial conditions are given byx(0) = y(0) = z(0) = x(0) = y(0) = z(0) = 0, we get thesolution

x(t) =QME0 [cos (ByQt/M)− cos(ωt)]

M2ω2 −B2yQ

2(29)

y(t) =QE0 [ωt− sin(ωt)]

Mω2(30)

z(t) =QE0 [ByQ sin(ωt)−Mω sin (ByQt/M)]

B2yQ

2ω −M2ω3(31)

Defining the values of the constants as E0 = Q = M =By = 1, we get the trajectories shown in the figures 11and 12.

Figura 11: Trajectory of a charged particle subjectedto fields (20) and (21), with the parameters M =E0 = By = Q = 1, ω = 0.99 and initial conditionsx(0) = y(0) = z(0) = x(0) = y(0) = z(0) = 0, in atime interval [0, 40π].

Figura 12: Trajectory of a charged particle subjectedto fields (20) and (21), with the parameters M =E0 = By = Q = 1, ω = 1.1 and initial conditionsx(0) = y(0) = z(0) = x(0) = y(0) = z(0) = 0, in atime interval [0, 50π].

The solutions (29), (30) and (31), can be obtained inMathematica from the commands

10

F = {Fx, Fy, Fz}; r = {x[t], y[t], z[t]}; EF

= {Ex, Ey , Ez}; BF = {Bx, By, Bz}; v = D[r,

t];

F = Q*(EF + Cross[v, BF])

eq1 = F - M*D[r, {t, 2}] == 0 // Thread; eq1

// ColumnForm

initial = {x[0] == 0, x’[0] == 0, y[0] == 0,

y’[0] == 0, z[0] == 0, z’[0] == 0};

fields = Thread /@ {BF -> {0, By, 0}, EF ->

{E0*Cos[ω*t],E0*Sin[ω*t], 0}} // Flatten

eqs = Join[eq1 , inicial] /. feilds

DSolve[eqs , {x[t], y[t], z[t]}, t] //

Flatten // FullSimplify

From the solutions obtained, we used to build the si-mulation illustrated in the figure 11 The codes

values ={M-> 1,E0 -> 1,By -> 1,Q-> 1}

eqx[t_] := (E0 M Q (Cos[(By Q t)/M]-Cos[t ω]))/(-By^2 Q^2 + M^2 ω^2);eqy[t_] := (E0 Q (t ω - Sin[t ω]))/(M ω^2);eqz[t_] := (E0 Q (-M ω Sin[(By Q t)/M] + By

Q Sin[t ω]))/(By^2 Q^2 ω - M^2 ω^3);

ParametricPlot3D [{eqx[t], eqy[t], eqz[t]} /.

ω -> 0.99 /. valores , {t, 0, 20*Pi},

AxesLabel -> {"x", "y", "z"}, Mesh -> None ,

PlotRange -> All , PlotStyle -> {Red ,

Directive[EdgeForm []], Thickness [0.005] ,

Opacity [1]}, ImageSize -> Medium]

from which we can obtain the trajectory of the figure 12analogously, varying the value of ω.

5. ~E oscillating and ~B uniform, in two directions (xy)

Consider a charged particle with charge Q and massM , subjected to an oscillating electric field and a uniformmagnetic field, both in two directions,

~E = E0 cos(ωt)i+ E0 sin(ωt)j (32)

~B = Bxi+By j. (33)

Compared to the previous case, there has only beenan addition of one component Bx. However, this subtlechange is enough to greatly complicate the solution ofthe equations of motion. Therefore, this is a problemthat can be solved efficiently and much more practically

by using Mathematica software than by performing thealgebraic calculations by hand.

For this system, the resulting equations of motion aregiven by

Q [E0 cos(ωt)−By z]−Mx = 0 (34a)

Q [E0 sin(ωt) +Bxz]−My = 0 (34b)

Q [Byx−Bxy]−Mz = 0 (34c)

Although the system (34) of coupled ODEs is very la-borious to solve, there is a general analytical solution.Assuming that the initial conditions are x(0) = y(0) =z(0) = x(0) = y(0) = z(0) = 0, the general solution isobtained by means of the commands

F = {Fx, Fy, Fz}; r = {x[t], y[t], z[t]}; EF

= {Ex, Ey , Ez}; BF = {Bx, By, Bz}; v = D[r,

t];

F = Q*(EF + Cross[v, BF])

eq1 = F - M*D[r, {t, 2}] == 0 // Thread; eq1

// ColumnForm

initial = {x[0] == 0, x’[0] == 0, y[0] == 0,

y’[0] == 0, z[0] == 0, z’[0] == 0};

feilds = Thread /@ {BF -> {B_x , By, 0}, EF

-> {E0*Cos[ω*t],E0*Sin[ω*t], 0}} // Flatten

eqs = Join[eq1 , initial] /. feilds

DSolve[eqs , {x[t], y[t], z[t]}, t] //

Flatten // FullSimplify

Defining the constants A1, A2, A3, A4, A5, A6, k1 e k2as

A1 =1

Mω2[B2

x +B2y

] [Q2(B2

x +B2y

)−M2ω2

] (35)

A2 =

√B2

x +B2y√

−(B2

x +B2y

) (36)

A3 = Q2(B2

x +B2y

)−M2ω2 (37)

A4 =B2

xM3ω3

Q√B2

x +By2 (38)

A5 = BxByQ2(B2

x +B2y

)(39)

A6 =√B2

x +B2y (40)

k1 = −Q√−(Bx2 +By2)

M(41)

k2 =Q√B2

x +B2y

M(42)

the general solution is given in the form

11

x(t) =−A1E0Qek1t [cos (k2t)−A2 sin (k2t)]

[−BxA3(Bx +Byωt) +B2

yM2ω2 cos (k2t)

+(B2

x +B2y

)(BxQ−Mω)(BxQ+Mω) cos(ωt) +A4By sin(k7t) +A5 sin(ωt)

](43)

y(t) =A1E0Qek1t [cos (k2t)−A2 sin (k2t)]

[ByA3(Bx +Byωt) +BxByM

2ω2 cos (k2t)

−(B2

x +B2y

)(ByQ−Mω)(ByQ+Mω) sin(ωt) +A4Bx sin(k7t)−A5 cos(ωt)

](44)

z(t) =A1E0Qek1t

{√B2

x +B2y cos (k2t) +

√−(B2

x +B2y) sin (k2t)

}{−BxA6A3 −BxA6M

2ω2 cos (k2t)

+Q(B2

x +B2y

) [Q√B2

x +B2y(Bx cos(ωt) +By sin(ωt))−ByMω sin (k2t)

]}(45)

With the solutions (43), (44) and (45) at hand, wecan make computer simulations of possible trajectoriesperformed by the charged particle, as shown in the figure13.

Figura 13: Trajectory of a charged particle subjectedto fields (32) and (33), with the parameters M =E0 = Bx = Q = 1, ω = 2.2 and initial conditionsx(0) = y(0) = z(0) = x(0) = y(0) = z(0) = 0, in atime interval [0, 55π].

For the figure to be obtained, it is necessary to definethe value of the parameters of the equation,

values ={M->1,E0 ->1,Bx ->1,By ->1,Q->1}

followed by the ParameticPlot3d command with input onthe solutions obtained,

ParametricPlot3D [{eqx[t], eqy[t], eqz[t]} /.

ω -> 2 /. values , {t, 0, 20*Pi}, AxesLabel

-> {"x", "y", "z"}, Mesh -> None , PlotRange

-> All , PlotStyle -> {Red , Directive[

EdgeForm []], Thickness [0.005] , Opacity [1]},

ImageSize -> Medium]

6. ~E oscillating in two directions (xy) and ~B uniform inthree (xyz)

Considering the case treated in the example III A 6,an oscillating electric and uniform magnetic field in twodirections, differing only in the constant magnetic fieldsuperposition on the z axis, i.e,

~E = E0 cos(ωt)i+ E0 sin(ωt)j (46)

~B = Bxi+By j +Bz k, (47)

a forca de Lorentz nos vela as equacoes de movimento

Q [E0 cos(ωt) +Bz y −By z]−Mx = 0 (48a)

Q [E0 sin(ωt)−Bzx+Bxz]−My = 0 (48b)

Q [Byx−Bxy]−Mz = 0 (48c)

Mathematica finds the general solution of the system(48) with the codes

F = {Fx, Fy, Fz}; r = {x[t], y[t], z[t]}; EF

= {Ex, Ey , Ez}; BF = {Bx, By, Bz}; v = D[r,

t];

F = Q*(EF + Cross[v, BF])

eq1 = F - M*D[r, {t, 2}] == 0 // Thread; eq1

// ColumnForm

initial = {x[0] == 0, x’[0] == 0, y[0] == 0,

y’[0] == 0, z[0] == 0, z’[0] == 0};

feilds = Thread /@ {BF -> {B_x , By, Bz}, EF

-> {E0*Cos[ω*t],E0*Sin[ω*t], 0}} // Flatten

eqs = Join[eq1 , initial] /. feilds

DSolve[eqs , {x[t], y[t], z[t]}, t] //

Flatten // FullSimplify

12

where, we assume that the initial conditions are x(0) = y(0) = z(0) = x(0) = y(0) = z(0) = 0.Defining the constants

A1 =1

2MQω2[B2

x +B2y +B2

z

]5/2 [Q2(B2

x +B2y +B2

z

)−M2ω2

] (49)

A2 =√B2

x +B2y +B2

z

[Q2(B2

x +B2y +B2

z

)−M2ω2

](50)

A3 = B2x +B2

y +B2z (51)

A4 = +2M2ω2√B2

x +B2y +B2

z (52)

A5 = M2ω2(−A3EzQ2 −BxE0MQω + EzM

2ω2) (53)

A6 = 2ByM2ω2√B2

x +B2y +B2

z

[A3(BxE0 +BzEz)Q2 −BzEzM

2ω2]

(54)

A7 = M2ω2[(B2

x +B2y +B2

z )(−BzE0 +BxEz)Q2 + (B2x +B2

z )E0MQω −BxEzM2ω2]

(55)

A8 = A3/23 E0Q

2[B2

yQ2 +Mω(BzQ−Mω)

](56)

A9 = 2M2ω2{A3Q

2[−BxBzE0 + (A3 −B2

z )Ez

]+BxA3E0MQω − (A3 −B2

z )EzM2ω2}

(57)

A10 =1

[2A23ω

2(A3MQ3 −M3Qω2)](58)

k1 =√

A3Q/M (59)

k2 = Q√

B2x + B2

y + B2z/m (60)

we can write the solution in the form

x(t) =A1

{2A2

[2E0Q

2B2x + 2A3E0Qω(BxByQt+BzM) + Ezω

2(BxBzQ

2A3t2 − 2ByMQA3t− 2BxBzM

2) ]

+A4 cos(k1t)

[A3Q

2(BxBzEz − E0B

2y − E0B

2z

)+A3BzE0MQω −BxBzEzM

2ω2

]− 2A

5/23 E0B

2xQ

4

− 2A5/23 E0Q

2Mω(BzQ−Mω) cos(ωt)− 2ByA3

[A5 sin(k1t) +BxE0A

2/33 Q4 sin(ωt)

]}(61)

y(t) =A1

{A2

[2BxByA3E0Q

2 + 2A3E0Q2B2

yωt+ Ezω2(−2ByBzM

2 + 2BxA3MQt+ByBzA3Q2t2) ]

+A6 cos(k1t)− 2A3

[BxByA

3/23 E0Q

4 cos(ωt) +A7

]sin(k1t) +A8 sin(ωt)

}(62)

z(t) =A10

{[A3Q

2 −M2ω2

][2BxBzA3E0Q

2 − 2A3E0Q(BxM −ByBzQt)ω + 2Ez(B2x +B2

y)M2 + EzB2zA3Q

2t2ω2

]−A9 cos(k2t)− 2E0QA

1/23

[BxA

3/23 Q2(BzQ−Mω) cos(ωt) +ByM

2ω2(A3Q−BzMω) sin(k2t)

+ByA3/23 Q2(BzQ−Mω) sin(ωt)

]}(63)

In order to get the simulation, we will set the parame-ters M = E0 = Bx = By = Bz = Q = 1. This can bedone in Mathematica as follows,

values ={M->1,E0 ->1,Bx ->1,By ->1,Q->1}

Using ParametricPlot3d, we can call the solutions (61),(62) and (63), to generate the trajectory of the particle,by means of the commands

ParametricPlot3D [{eqx[t], eqy[t], eqz[t]} /.

ω -> 2 /. values , {t, 0, 20*Pi}, AxesLabel

-> {"x", "y", "z"}, Mesh -> None , PlotRange

-> All , PlotStyle -> {Red , Directive[

EdgeForm []], Thickness [0.005] , Opacity

[1]}, ImageSize -> Medium]

The result is the trajectory illustrated in the figure 14.

13

Figura 14: Trajectory of a charged particle subjectedto fields (46) and (47), with the parameters M =E0 = Bx = By = Bz = Q = 1, ω = 15 and initialconditions x(0) = y(0) = z(0) = x(0) = y(0) =z(0) = 0, in a time interval [0, 5π].

7. Simulation for an oscillating electric field

We can simulate an oscillating electric field of the casein the figure 10, where M = E0 = Bx = Q = 1 andω = 0.51. Through the commands

x[t_] := -((E0 q (-1 + Cos[t ω]))/(m ω^2));y[t_] := (E0 m (-m ω Sin[(Bx q t)/m] + Bx q Sin[

t ω]))/( Bx^3 q^2 - Bx m^2 ω^2);z[t_] := (E0(-Bx^2q^2+m^2ω^2-m^2ω^2Cos[(Bx q t)/

m]+

Bx^2 q^2 Cos[tω]))/(Bx^3 q^2 ω-Bx m^2 ω^3);

m = 1; q = 1; E0 = 1; Bx = 1;

tRange=Range [0.001 ,100*Pi ,100*Pi /500];ω=0.51;For[k = 1, k <= Length[tRange], k++, tk=tRange [[

k]]; xk=x[tk];yk=y[tk];zk=z[tk];

movieVector[k]=Show[Graphics3D [{

AbsolutePointSize [15], Green , Point [{xk , yk,

zk}]},

PlotRange -> {{-0.2, 8.2}, {-2.2, 2.2},

{-5.7, 1.7}}] ,

ParametricPlot3D [{x[t], y[t], z[t]}, {t, 0,

tk}, PlotStyle -> Red ,

PlotRange -> {{-0.2, 8.2}, {-2.2, 2.2},

{-5.7, 1.7}}] ,

AxesLabel -> {"x", "y", "z"},

PlotLabel -> StringJoin ["t=", ToString[N[tk

]]]]];

M = Table[movieVector[k], {k, 1, 500}];

fileName = "FilmeMathematica.avi"; Export[

fileName , M];

Directory []

The resulting simulation can be seen in the followingvideo https://youtu.be/nlUoLaV4ECc.

IV. CONCLUSION

In this paper we propose the use of Mathematica soft-ware to solve ordinary differential equations, and to re-present the trajectories of charged particles subjected tooscillating electric and constant magnetic fields.

We begin by showing the use of Mathematica softwarein solving the exact differential equation of the simpleharmonic oscillator, and the graphical representation ofthis solution. Then we show a numerical solution of thesimple pendulum, also showing the graphical representa-tion of that solution and its phase space.

Finally, we show how to use Mathematica software toconstruct the differential equations of Newton’s secondlaw of motion, which is the Lorentz force, defining po-sition vector, velocity, electric and magnetic fields, andthus solve the equations of motion, calling the solutionsto represent them graphically, as trajectories of the char-ged particle.

We end by showing how to build a video simulation ofthe charged particle trajectory representation.

We believe that this article serves as an introduction tosymbolic algebraic computation, for use in topics that aremore difficult to obtain non-analytic solutions, in severalareas of Physics and Mathematics. The generalization ofits use to solve partial differential equations, and nonli-near equations is possible.

V. ACKNOWLEDGMENTS

The authors thank CNPq for the PIBIC scholarshipand partial financing of this work.

14

[1] Harris, J.M. Utilization of computer technology by te-achers at Carl Schurz High School, a Chicago pu-blic school. Doctor of Education thesis, NorthernIllinois University. Retrieved April 5, 2021 fromhttps://www.learntechlib.org/p/116133/.

[2] A. R. Booker and A. V. Sutherland, arXiv:2007.01209.

[3] Jyoti Bhalla, Journal of Education and Training Studie1, 2 (2013).

[4] D. W. da M. Pastana and M. E. Rodigues, Rev. Bras.Ens. Fis. 34 (2021).

[5] Eylon, Bat-Sheva, et al. Journal of Science Educationand Technology, p. 93–110. JSTOR 5, 2 (1996).

[6] Luciano J. B. Quaresma e M. E. Rodrigues, Rev. Bras.Ensino Fıs. 41, 2 (2019).

[7] H. R. E. H. Bouchekara, H. Allag and M. Boucherma,International Journal of Electromagnetics and Applicati-ons 1, 1 (2011).

[8] J. D. Jackson, Classical Electrodynamics (Wiley, NewJersy, 1999), p. 808, 3ª ed.

[9] D. Crowe and H. Zand, Computers & Education 35, 2,pages 95-121 (2000).

[10] L. B. Challoo, R. L. Marshall and I. L. Marshall, NATIO-NAL FORUM OF TEACHER EDUCATION JOURNAL14, 3 (2004).

[11] K. R. Symon, Mechanics (Addison Wesley, Boston,1971), p. 639, 3ª ed.

[12] R. P. Feynman, F. B. Morinigo, W. G. Wagner and D.Pines, Feynman lectures on gravitation (Addison-Wesley,Boston, 1995), volume 1.

[13] H. Goldstein, C. P. Poole, J. L. Safko, Classical mecha-nics (Addison-Wesley, Boston, 2000), 3. ed.

[14] D. Crowe and H. Zand, Computers & Education 37, 3-4,pages 317-344 (2001).

[15] D. Crowe and H. Zand, Procedia – Social and BehavioralSciences 47, pages 1058-1063 (2012).

[16] A. A. Fidalgo-Neto, A. J. C. Tornaghi, R. M. S. Meirelles,F. F. Bercot, L. L. Xavier, M. F. A. Castro, L. A. Alves,Computers & Education 53, 3, pages 677-685 (2009).

[17] A. Jimoyiannis, V. Komis, Computers & Education 36,pages 183-204, (2001).

[18] S. T. Thornton , J. B. Marion, Cassical Dynamics ofParticles and Systems. (Brooks Cole, California, 2003),5ª ed.

[19] Sum of three cubes for 42 finally solved – usingreal life planetary computer. Disponıvel em:http://www.bristol.ac.uk/news/2019/september/sum-of-three-cubes-.html, accessed on 04/05/2021