Skip to main content

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


Comments

Post a Comment

Popular posts from this blog

functions - Get leading series expansion term?

Given a function f[x] , I would like to have a function leadingSeries that returns just the leading term in the series around x=0 . For example: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x)] x and leadingSeries[(1/x + 2 + (1 - 1/x^3)/4)/(4 + x)] -(1/(16 x^3)) Is there such a function in Mathematica? Or maybe one can implement it efficiently? EDIT I finally went with the following implementation, based on Carl Woll 's answer: lds[ex_,x_]:=( (ex/.x->(x+O[x]^2))/.SeriesData[U_,Z_,L_List,Mi_,Ma_,De_]:>SeriesData[U,Z,{L[[1]]},Mi,Mi+1,De]//Quiet//Normal) The advantage is, that this one also properly works with functions whose leading term is a constant: lds[Exp[x],x] 1 Answer Update 1 Updated to eliminate SeriesData and to not return additional terms Perhaps you could use: leadingSeries[expr_, x_] := Normal[expr /. x->(x+O[x]^2) /. a_List :> Take[a, 1]] Then for your examples: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x), x] leadingSeries[Exp[x], x] leadingSeries[(1/x + 2 + (1 - 1/x...
Post a Comment
Read more

How to thread a list

I have data in format data = {{a1, a2}, {b1, b2}, {c1, c2}, {d1, d2}} Tableform: I want to thread it to : tdata = {{{a1, b1}, {a2, b2}}, {{a1, c1}, {a2, c2}}, {{a1, d1}, {a2, d2}}} Tableform: And I would like to do better then pseudofunction[n_] := Transpose[{data2[[1]], data2[[n]]}]; SetAttributes[pseudofunction, Listable]; Range[2, 4] // pseudofunction Here is my benchmark data, where data3 is normal sample of real data. data3 = Drop[ExcelWorkBook[[Column1 ;; Column4]], None, 1]; data2 = {a #, b #, c #, d #} & /@ Range[1, 10^5]; data = RandomReal[{0, 1}, {10^6, 4}]; Here is my benchmark code kptnw[list_] := Transpose[{Table[First@#, {Length@# - 1}], Rest@#}, {3, 1, 2}] &@list kptnw2[list_] := Transpose[{ConstantArray[First@#, Length@# - 1], Rest@#}, {3, 1, 2}] &@list OleksandrR[list_] := Flatten[Outer[List, List@First[list], Rest[list], 1], {{2}, {1, 4}}] paradox2[list_] := Partition[Riffle[list[[1]], #], 2] & /@ Drop[list, 1] RM[list_] := FoldList[Transpose[{First@li...
Post a Comment
Read more

front end - keyboard shortcut to invoke Insert new matrix

I frequently need to type in some matrices, and the menu command Insert > Table/Matrix > New... allows matrices with lines drawn between columns and rows, which is very helpful. I would like to make a keyboard shortcut for it, but cannot find the relevant frontend token command (4209405) for it. Since the FullForm[] and InputForm[] of matrices with lines drawn between rows and columns is the same as those without lines, it's hard to do this via 3rd party system-wide text expanders (e.g. autohotkey or atext on mac). How does one assign a keyboard shortcut for the menu item Insert > Table/Matrix > New... , preferably using only mathematica? Thanks! Answer In the MenuSetup.tr (for linux located in the $InstallationDirectory/SystemFiles/FrontEnd/TextResources/X/ directory), I changed the line MenuItem["&New...", "CreateGridBoxDialog"] to read MenuItem["&New...", "CreateGridBoxDialog", MenuKey["m", Modifiers-...
Post a Comment
Read more