File:Chain of pendula.gif
Summary
| Description |
English: If instead of a double pendulum you chain many together, they behave like a chain. If there is no friction or restriction on curvature, the system start showing its chaotic nature after a while. |
| Date | |
| Source | https://twitter.com/j_bertolotti/status/1346753297223458817 |
| Author | Jacopo Bertolotti |
| Permission (Reusing this file) |
https://twitter.com/j_bertolotti/status/1030470604418428929 |
Mathematica 12.0 code
n = 10;
vars = Table[\[Theta][j, t], {j, 1, n}];
coord = Accumulate@Table[r[j, t] = {Sin[vars[[j]] ], Cos[vars[[j]] ]}, {j, 1, n}];
norm[v_] := Sqrt[Sum[v[[j]]^2, {j, 1, Dimensions[v][[1]]}]];
L = FullSimplify[(Total[(norm /@ D[coord, t])^2] - Total@coord[[All, 2]])];
eqs = FullSimplify@Table[D[D[L, \!\(\*SuperscriptBox[\(\[Theta]\), TagBox[RowBox[{"(", RowBox[{"0", ",", "1"}], ")"}],Derivative],MultilineFunction->None]\)[j, t]], t] - D[L, \[Theta][j, t]] == 0, {j,1, n}];
initialcond01 = Table[\[Theta][j, 0] == \[Pi] - \[Pi]/2.5 - RandomReal[{-10^-6, 10^-6}], {j, 1, n}];
initialcond02 = Table[\[Theta][j, 0] == \[Pi] - \[Pi]/2.5 - RandomReal[{-10^-6, 10^-6}], {j, 1, n}];
initialcond03 = Table[\[Theta][j, 0] == \[Pi] - \[Pi]/2.5 - RandomReal[{-10^-6, 10^-6}], {j, 1, n}];
initialcond2 = Table[\!\(\*SuperscriptBox[\(\[Theta]\), TagBox[RowBox[{"(", RowBox[{"0", ",", "1"}], ")"}],Derivative], MultilineFunction->None]\)[j, 0] == 0, {j, 1, n}];
sol1 = NDSolve[Join[eqs, initialcond01, initialcond2], vars, {t, 0, 500}, Method -> {"EquationSimplification" -> "Residual"}];
sol2 = NDSolve[Join[eqs, initialcond02, initialcond2], vars, {t, 0, 500}, Method -> {"EquationSimplification" -> "Residual"}];
sol3 = NDSolve[Join[eqs, initialcond03, initialcond2], vars, {t, 0, 500}, Method -> {"EquationSimplification" -> "Residual"}];
frames = Table[
Graphics3D[{
Thickness[0.005],
Cyan,
Sphere[{0, 0, 0}, 0.2],
Table[ Sphere[ Join[Evaluate[((coord /. sol1)[[1, j]]) /. {t -> \[Tau]} ], {0}], 0.2], {j, 1, n}],
Line[Join[{{0, 0, 0}}, Table[Join[Evaluate[((coord /. sol1)[[1, j]]) /. {t -> \[Tau]} ], {0}], {j, 1, n}]]]
,
Orange,
Sphere[{0, 0, 0}, 0.2],
Table[Sphere[Join[Evaluate[((coord /. sol2)[[1, j]]) /. {t -> \[Tau]} ], {0}], 0.21], {j, 1, n}],
Line[Join[{{0, 0, 0}}, Table[Join[Evaluate[((coord /. sol2)[[1, j]]) /. {t -> \[Tau]} ], {0}], {j, 1, n}]]]
,
Purple,
Sphere[{0, 0, 0}, 0.2],
Table[Sphere[Join[Evaluate[((coord /. sol3)[[1, j]]) /. {t -> \[Tau]} ], {0}], 0.22], {j, 1, n}],
Line[Join[{{0, 0, 0}}, Table[Join[Evaluate[((coord /. sol3)[[1, j]]) /. {t -> \[Tau]} ], {0}], {j, 1, n}]]]
}, PlotRange -> {{-n - 1, n + 1}, {-n - 1, n/2}, {-1, 1}}, Background -> Black, ViewPoint -> Above, Boxed -> False]
, {\[Tau], 0, 300, 1}];
ListAnimate[frames]
Licensing
I, the copyright holder of this work, hereby publish it under the following license:
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This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication. |
| The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.
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