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differential equations - Using NDSolve to find particle trajectory


I'm trying to simulate a particle in an electric and magnetic fields, but numerically instead of analytically. This is basically solving the equation


q⋅(p′×B)+q⋅E=mp″,


where p(t) is the position in (x,y,z) coordinates.


After viewing a few topics on this site, I've got a good idea on how to get the solution using NDSolve, but my program gets stuck, and doesn't come up with anything.


b = {1, 0, 0};
e = {0, 0, 1};
q = 1;

m = 1;

sol = NDSolve[ {q*e + q*Cross[D[pos[t], t], b] == m D[pos[t], {t, 2}],
     pos[0] == {0, 0, 0}, (D[pos[t], t] /. t -> 0) == {0, 0, 0}}, 
   pos, {t, 0, 1}];
ParametricPlot3D[Evaluate[pos[t] /. sol], {t, 0, 1}];

It is also worth mentioning that if you remove the qâ‹…E term, the calculation is finished, but nothing shows up in the plot.



Answer



The main problem is that your pos is not seen as a 3D vector.



The cross product is therefore interpreted as a scalar:


q*Cross[D[pos[t], t], b]

Mathematica graphics


when adding this to the vector q.e this 'scalar' term is added to each of the vector components:


q*e + q*Cross[D[pos[t], t], b]

Mathematica graphics


This won't work, instead do:


b = {1, 0, 0};

e = {0, 0, 1};
q = 1;
m = 1;

Define pos as a 3D vector. Also take more time than a single second:


ClearAll[pos]
pos[t_] = {px[t], py[t], pz[t]};
sol = NDSolve[
  {
   q*e + q*Cross[D[pos[t], t], b] == m D[pos[t], {t, 2}],

   pos[0] == {0, 0, 0},
   (D[pos[t], t] /. t -> 0) == {0, 0, 0}
   }, pos[t], {t, 0, 20}]

ParametricPlot3D[Evaluate[pos[t] /. sol], {t, 0, 20}]

Mathematica graphics


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