Jump to content

File:Closed orbits.gif

Page contents not supported in other languages.
This is a file from the Wikimedia Commons
From Wikipedia, the free encyclopedia

Closed_orbits.gif (592 × 483 pixels, file size: 6.52 MB, MIME type: image/gif, looped, 300 frames, 30 s)

Summary

Description
English: 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:
Creative Commons CC-Zero 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.

Captions

Central forces proportional to 1/r² and r are the only ones that guarantee that all bound orbits are closed.

Items portrayed in this file

depicts

7 April 2020

image/gif

479e909b7cbc3b46fa9f179445e1993e5aea35e2

6,835,038 byte

30.000000000000156 second

483 pixel

592 pixel

File history

Click on a date/time to view the file as it appeared at that time.

Date/TimeThumbnailDimensionsUserComment
current12:37, 8 April 2020Thumbnail for version as of 12:37, 8 April 2020592 × 483 (6.52 MB)BertoUploaded own work with UploadWizard

The following page uses this file:

Global file usage

The following other wikis use this file:

Metadata