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Summary
| Description |
Central forces that decay as 1/r² are special, as they guarantee that all bound orbits are going to be closed (Bertrand's theorem).
Small changes in the power will lead to significantly different kind of orbits. |
| Date | |
| Source | https://twitter.com/j_bertolotti/status/1247542284616269826 |
| Author | Jacopo Bertolotti |
| Permission (Reusing this file) |
https://twitter.com/j_bertolotti/status/1030470604418428929 |
Mathematica 12.0 code
ep = {0, 0};
me = 5;
mp = {{0, 5}, {0, 5}, {0, 5}};
acc = {me (ep - mp[[1]])/(Norm[ep - mp[[1]]]^1 Norm[ep - mp[[1]]]^2), me (ep - mp[[2]])/(Norm[ep - mp[[2]]]^1 Norm[ep - mp[[2]]]^1.9), me (ep - mp[[3]])/(Norm[ep - mp[[3]]]^1 Norm[ep - mp[[3]]]^2.1)};
mv = mv = {{Sqrt[Abs[Norm[acc[[1]] ] Norm[mp[[1]] - ep] ]], 0.2}, {Sqrt[Abs[Norm[acc[[1]] ] Norm[mp[[1]] - ep] ]], 0.2}, {Sqrt[Abs[Norm[acc[[1]] ] Norm[mp[[1]] - ep] ]], 0.2}};
dt = 0.2;
mpold = mp;
mp = mpold + mv dt + acc/2 dt^2;
evo = Reap[Do[
acc = {me (ep - mp[[1]])/(Norm[ep - mp[[1]]]^1 Norm[ep - mp[[1]]]^2),
me (ep - mp[[2]])/(Norm[ep - mp[[2]]]^1 Norm[ep - mp[[2]]]^1.9),
me (ep - mp[[3]])/(Norm[ep - mp[[3]]]^1 Norm[ep - mp[[3]]]^2.1)};
mpoldold = mpold;
mpold = mp;
mp = 2 mpold - mpoldold + acc dt^2;
Sow[mp];
, {1500}];][[2, 1]];
plots = Table[
Legended[
Graphics[{Gray, Disk[ep, 0.1 ],
Purple, Disk[evo[[j, 2]], 0.5 ], Line[evo[[1 ;; j, 2]] ]
,
Orange, Disk[evo[[j, 3]], 0.5 ], Line[evo[[1 ;; j, 3]] ]
,
Black, Disk[evo[[j, 1]], 0.5 ], Line[evo[[1 ;; j, 1]] ]
},
PlotRange -> {{-10, 10}, {-10, 10}}, Frame -> False], LineLegend[{Black, Purple, Orange}, {"F\[Proportional]\!\(\*FractionBox[\(1\), \SuperscriptBox[\(r\), \(2\)]]\)", "F\[Proportional]\!\(\*FractionBox[\(1\), SuperscriptBox[\(r\), \(1.9\)]]\)", "F\[Proportional]\!\(\*FractionBox[\(1\), SuperscriptBox[\(r\), \(2.1\)]]\)"}] ]
, {j, 1, Dimensions[evo][[1]]}];
ListAnimate[plots[[1 ;; -1 ;; 5]] ]
Licensing
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.
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File history
Click on a date/time to view the file as it appeared at that time.
| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 12:37, 8 April 2020 | | 592 × 483 (6.52 MB) | Berto | Uploaded own work with UploadWizard |
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