problem17
17. Montgomery’s eight. (From Ruina/Pratap). Three equal masses, say \(m = 1\), are attracted by an inverse-square gravity law with \(G = 1\). That is, each mass is attracted to the other by \(F = G m_1 m_2 / r^2\) where \(r\) is the distance between them. Use these unusual and special initial positions:
\[ \begin{align*} (x1, y1) &= (−0.97000436, 0.24308753) \\ (x2, y2) &= (−x1, −y1) \\ (x3, y3) &= (0, 0) \end{align*} \]
and initial velocities
$$ \[\begin{align*} (vx3, vy3) &= (0.93240737, 0.86473146) (vx1, vy1) &= −(vx3, vy3)/2 (vx2, vy2) &= −(vx3, vy3)/2. \end{align*}\]
For each of the problems below show accurate computer plots and explain any curiosities.
- Use computer integration to find and plot the motions of the particles. Plot each with a different color. Run the program for 2.1 time units.
- Same as above, but run for 10 time units.
- Same as above, but change the initial conditions slightly.
- Same as above, but change the initial conditions more and run for a much longer time.
[Aside: This was discovered by both Richard Montgomery (Santa Cruz Math department) and also by ex-Cornell Physics PhD student Chris Moore (now at Santa FeInstitute), independently. And, I know both of them, independently.]
