DSolve; Lorentz Force equations of motion

  • Thread starter KleZMeR
  • Start date
  • #1
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Hi,

I am trying to use DSolve in Mathematica 6.0.1.0, in order to find equations of motion (x[t], y[t]) from second-order differential equations. I have looked through much documentation, and attempted numerous codes to figure this out, but get many errors or the same answer every time. This is my first Mathematica course. I derived the second order equations by hand, which is found in many books.
I do not see how the equations of motion I am to show, x[t], y[t], are produced by DSolve.
Below is stated problem and attempt.

Electron injected with initial velocity Vo into crossed field; E=Ey, B=Bz,
and: w=omega, Vd=(Ey/Bz)
Also given: w=1, Vox=Voy=2, Vd=1, (I do not know why these are given, unless they are initial conditions, and even so, they still do not change my outcome)

a) Use F = q (E + V x B) and show by hand that motion is governed by these equations (I did this):

x''[t] + (w^2)x = (w^2)*Vd*t - w*Voy

y''[t] + (w^2)y = -(w^2)*Vd*t + w*Vox



b) Use DSolve to show that the solutions to these equations are:

x[t] = (1/w)*(Vox - Vd)*sin(wt) + (Voy/w)*(cos(wt) - 1) + Vd*t

y[t] = (Voy/w)*sin(wt) - (1/w)*(Vox - Vd)*(cos(wt) - 1)




So I am having trouble with part b)... I am sure that I am missing much in my attempt at solving it, but all other inputs either give me an error, or do not change the output, any help or suggestions would be appreciated, thanks.
 

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Answers and Replies

  • #2
609
0
DSolve[x''[t] + Subscript[\[Omega], c]^2 x[t] ==
Subscript[\[Omega], c]^2 Subscript[v, d]*t -
Subscript[\[Omega], c]*Subscript[v, 0], x[t], t]


i did it and it works

Try restarting the kernel (Evaluation - Quit Kernal - Local (or other)) and just running that one line.
 
  • #3
127
1
Awsome! Thank you so much K.J. !!!

I would also like to replace the two constants of integration(C[1], C[2]) with two chosen variables

If I include this in DSolve:

{C[1] -> Vox, C[2] -> Voy},

it returns that I can not use these for variables.

I have also found a GeneratedParameters-> function, but it does not work either,.. and if I'm correct, the GeneratedParameters-> Module{C[1], C[2]..&} function is only to ensure that C[] values are all unique, and does not change their representation.

And last, these two equations of motion, with initial conditions, don't seem to give a parametric plot, just an empty axis. I am attaching this attempt at the parametric plot. This form does plot a different set of equations, so perhaps it is the equation representation,

Again, any help would be appreciated
Thanks
 

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