(*Staring positions in a triangle*)
x10 = -1;
y10 = -1;
x20 = 1;
y20 = -1;
x30 = 1;
y30 = 1;
(*Initial total momentum is zero, so the center of mass does not \
drift away*)
vx10 = 0.2;
vy10 = 0;
vx20 = -0.1;
vy20 = 0;
vx30 = 0;
vy30 = -0.1;
(*max time the system evolves (in arbitrary units)*)
T = 40;
(*All three bodies have the same mass*)
m1 = 1;
m2 = 1;
m3 = 1;
(*Setting up of the equations copied from \
http://demonstrations.wolfram.com/PlanarThreeBodyProblem/
There are more elegant and compact ways of doing this, but I wasn't \
interested in optimizing the code.*)
nds = NDSolve[
   {x1'[t] == vx1[t], y1'[t] == vy1[t],
    x2'[t] == vx2[t], y2'[t] == vy2[t],
    x3'[t] == vx3[t], y3'[t] == vy3[t],
    m1 vx1'[t] == -((
       m1 m2 (x1[t] - 
          x2[t]))/((x1[t] - x2[t])^2 + (y1[t] - y2[t])^2)^(3/2)) - (
      m1 m3 (x1[t] - x3[t]))/((x1[t] - x3[t])^2 + (y1[t] - y3[t])^2)^(
      3/2), m1 vy1'[t] == -((
       m1 m2 (y1[t] - 
          y2[t]))/((x1[t] - x2[t])^2 + (y1[t] - y2[t])^2)^(3/2)) - (
      m1 m3 (y1[t] - y3[t]))/((x1[t] - x3[t])^2 + (y1[t] - y3[t])^2)^(
      3/2), m2 vx2'[t] == (
      m1 m2 (x1[t] - x2[t]))/((x1[t] - x2[t])^2 + (y1[t] - y2[t])^2)^(
      3/2) - (m2 m3 (x2[t] - 
         x3[t]))/((x2[t] - x3[t])^2 + (y2[t] - y3[t])^2)^(3/2), 
    m2 vy2'[t] == (
      m1 m2 (y1[t] - y2[t]))/((x1[t] - x2[t])^2 + (y1[t] - y2[t])^2)^(
      3/2) - (
      m2 m3 (y2[t] - y3[t]))/((x2[t] - x3[t])^2 + (y2[t] - y3[t])^2)^(
      3/2), m3 vx3'[t] == (
      m1 m3 (x1[t] - x3[t]))/((x1[t] - x3[t])^2 + (y1[t] - y3[t])^2)^(
      3/2) + (m2 m3 (x2[t] - 
         x3[t]))/((x2[t] - x3[t])^2 + (y2[t] - y3[t])^2)^(3/2), 
    m3 vy3'[t] == (
      m1 m3 (y1[t] - y3[t]))/((x1[t] - x3[t])^2 + (y1[t] - y3[t])^2)^(
      3/2) + (m2 m3 (y2[t] - 
         y3[t]))/((x2[t] - x3[t])^2 + (y2[t] - y3[t])^2)^(3/2),
    x1[0] == x10, y1[0] == y10, x2[0] == x20, y2[0] == y20, 
    x3[0] == x30, y3[0] == y30,
    vx1[0] == vx10, vy1[0] == vy10, vx2[0] == vx20, vy2[0] == vy20, 
    vx3[0] == vx30, vy3[0] == vy30},
   {x1, x2, x3, y1, y2, y3, vx1, vx2, vx3, vy1, vy2, vy3}, {t, 0, 
    T}];
funsToPlot = {{x1[t], y1[t]}, {x2[t], y2[t]}, {x3[t], y3[t]}} /. 
   nds[[1]];
evo = Table[funsToPlot /. {t -> j}, {t, 0, T, 0.01}];
dim = Dimensions[evo][[1]];
(*For the elastic force case I used a Verlet integration, as this \
case is numerically very stable.*)
np = 3;
k0 = 1;
(*Same initial condition as the gravitational case*)

pos = {{x10, y10}, {x20, y20}, {x30, y30}};
v0 = {{vx10, vy10}, {vx20, vy20}, {vx30, vy30}};
acc = Table[
   Sum[If[j == k, 0, -k0 (pos[[j]] - pos[[k]])], {k, 1, np}], {j, 1, 
    np}];
dt = 0.005;
posold = pos;
pos = posold + v0 dt + acc/2 dt^2;
range = 5;

evoe = Reap[Do[
      acc = 
       Table[Sum[
         If[j == k, 0, -k0 (pos[[j]] - pos[[k]])], {k, 1, np}], {j, 1,
          np}];
      posoldold = posold;
      posold = pos;
      pos = 2 posold - posoldold + acc dt^2;
      Sow[pos];
      , dim];][[2, 1]];
plots = Table[
   GraphicsRow[{
     Show[
      ListPlot[{evo[[All, 1]][[1 ;; j]], evo[[All, 2]][[1 ;; j]], 
        evo[[All, 3]][[1 ;; j]]}, PlotStyle -> {Purple, Orange, Cyan},
        PlotRange -> {{-range, range}, {-range, range}}, 
       Joined -> True, Axes -> False, 
       PlotLabel -> "Gravitational 3-body problem", 
       LabelStyle -> {Bold, Black}],
      Graphics[{PointSize[0.03], Purple, Point[evo[[All, 1]][[j]]], 
        Orange, Point[evo[[All, 2]][[j]]], Cyan, 
        Point[evo[[All, 3]][[j]]]} , 
       PlotRange -> {{-range, range}, {-range, range}}], 
      ImageSize -> Medium
      ]
     ,
     Show[
      ListPlot[{evoe[[All, 1]][[1 ;; j]], evoe[[All, 2]][[1 ;; j]], 
        evoe[[All, 3]][[1 ;; j]]}, 
       PlotStyle -> {Purple, Orange, Cyan}, 
       PlotRange -> {{-range, range}, {-range, range}}, 
       Joined -> True, Axes -> False, 
       PlotLabel -> "Elastic 3-body problem", 
       LabelStyle -> {Bold, Black}],
      Graphics[{PointSize[0.03], Purple, Point[evoe[[All, 1]][[j]]], 
        Orange, Point[evoe[[All, 2]][[j]]], Cyan, 
        Point[evoe[[All, 3]][[j]]]} , 
       PlotRange -> {{-range, range}, {-range, range}}], 
      ImageSize -> Medium
      ]
     }], {j, 1, dim, 20}];
ListAnimate[plots]

I, the copyright holder of this work, hereby publish it under the following licence:

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 neighbouring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission. http://creativecommons.org/publicdomain/zero/1.0/deed.enCC0Creative Commons Zero, Public Domain Dedicationfalsefalse

This file, which was originally posted to https://twitter.com/j_bertolotti/status/1044947721696808961, was reviewed on 19 October 2018 by reviewer Ronhjones, who confirmed that it was available there under the stated license on that date.