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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


$$q \cdot \left(p'\times B\right) + q\cdot E = m p'',$$


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 $q\cdot E$ 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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