File:Lagrangian KleinBottle.gif
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Lagrangian_KleinBottle.gif (451 × 576 pixels, file size: 10.78 MB, MIME type: image/gif, looped, 200 frames, 20 s)
Captions
Captions
Point mass sliding on a Klein bottle
Summary[edit]
| Description |
English: A point mass sliding without friction on a Klein bottle under the effect of gravity.
(Starting with a small velocity) |
| Date | |
| Source | https://twitter.com/j_bertolotti/status/1259832002649391105 |
| Author | Jacopo Bertolotti |
| Permission (Reusing this file) |
https://twitter.com/j_bertolotti/status/1030470604418428929 |
Mathematica 12.0 code[edit]
x[t_] := -(2/15) Cos[u[t]] (3 Cos[v[t]] - 30 Sin[u[t]] + 90 Cos[u[t]]^4 Sin[u[t]] - 60 Cos[u[t]]^6 Sin[u[t]] + 5 Cos[u[t]] Cos[v[t]] Sin[u[t]])
y[t_] := 2/15 (3 + 5 Cos[u[t]] Sin[u[t]]) Sin[v[t]]
z[t_] := 1/15 Sin[u[t]] (3 Cos[v[t]] - 3 Cos[u[t]]^2 Cos[v[t]] - 48 Cos[u[t]]^4 Cos[v[t]] + 48 Cos[u[t]]^6 Cos[v[t]] - 60 Sin[u[t]] + 5 Cos[u[t]] Cos[v[t]] Sin[u[t]] - 5 Cos[u[t]]^3 Cos[v[t]] Sin[u[t]] -
80 Cos[u[t]]^5 Cos[v[t]] Sin[u[t]] + 80 Cos[u[t]]^7 Cos[v[t]] Sin[u[t]])
T[t_] := FullSimplify[m/2 (D[x[t], t]^2 + D[y[t], t]^2 + D[z[t], t]^2)]
V[t_] := m g z[t]
L = T[t] - V[t];
equ = D[D[L, u'[t]], t] - D[L, u[t]];
eqv = D[D[L, v'[t]], t] - D[L, v[t]];
m = 1; g = 1;
sol = NDSolve[{equ == 0, eqv == 0, u[0] == \[Pi], u'[0] == 0, v[0] == 0, v'[0] == 0.1}, {u[t], v[t]}, {t, 0, 100}, Method -> {"EquationSimplification" -> "Residual"}]
m =.; g =.;
p1 = Table[
Show[
ParametricPlot3D[{-(2/15) Cos[u] (3 Cos[v] - 30 Sin[u] + 90 Cos[u]^4 Sin[u] - 60 Cos[u]^6 Sin[u] + 5 Cos[u] Cos[v] Sin[u]),
2/15 (3 + 5 Cos[u] Sin[u]) Sin[v],
1/15 Sin[u] (3 Cos[v] - 3 Cos[u]^2 Cos[v] - 48 Cos[u]^4 Cos[v] + 48 Cos[u]^6 Cos[v] - 60 Sin[u] + 5 Cos[u] Cos[v] Sin[u] - 5 Cos[u]^3 Cos[v] Sin[u] - 80 Cos[u]^5 Cos[v] Sin[u] +
80 Cos[u]^7 Cos[v] Sin[u]) }
, {u, 0, \[Pi]}, {v, 0, 2 \[Pi]}, PlotStyle -> {Opacity[0.5]}, Boxed -> False, Axes -> False]
,
Graphics3D[{Black,
Sphere[Evaluate[{x[t], y[t], z[t]} /. sol][[1]] /. {t -> \[Tau]max}, 0.1]}]
,
ParametricPlot3D[
Evaluate[{x[t], y[t], z[t]} /. sol][[1]] /. {t -> \[Tau]}, {\[Tau], 0, \[Tau]max}, PlotStyle -> {Black, Thick, Dashed}]
]
, {\[Tau]max, 0.01, 100, 0.5}];
ListAnimate[p1]
Licensing[edit]
I, the copyright holder of this work, hereby publish it under the following license:
 
|
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.
|
File history
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 12:40, 12 May 2020 | | 451 × 576 (10.78 MB) | Berto (talk | contribs) | Uploaded own work with UploadWizard |
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