Chapter 6.3, Problem 16CP

A remarkable three-body figure-eight orbit was discovered by C. Moore in 1993. In this configuration, three bodies of equal mass chase one another along a single figure-eight loop. Set the masses to $m_1=m_2=m_3=1$

and gravity $g=1$

. (a) Adapt orbit. mto plot the trajectory with initial conditions $\left(x_1,y_1\right)=\left(-0.970,0.243\right),\left(x{\text{'}}_1,y{\text{'}}_1\right)=\left(-0.466,0.433\right),\left(x_2,y_2\right)=\left(-x_1,-y_1\right),\left(x{\text{'}}_2,y{\text{'}}_2\right)=\left(x{\text{'}}_1,y{\text{'}}_1\right)$ and $\left(x_3,y_3\right)=\left(0,0\right),\left(x{\text{'}}_2,y{\text{'}}_2\right)=\left(x{\text{'}}_3,y{\text{'}}_3\right)=\left(-2x{\text{'}}_1,-2y{\text{'}}_1\right)$ . (b)Are the trajectories sensitive to small changes in initial conditions? Investigate the effect of changing $x{\text{'}}_3$ by ${10}^{-k}$ for $1\le k\le 5$ . For each k,decide whether the figure-eight pattern persists, or a catastrophic change eventually occurs.