Similar to the previous step, an additional satellite was added, so that three bodies were now in motion relating to each other. This problem is very important in science, as even when the motion of all the bodies is in one plane the trajectories can turn out to be unpredictable. This unpredictability ends up coming from the sensitive dependence on the initial conditions. Basically this means that small shifts in the initial positions of the bodies, or the initial velocities of the bodies can have a major impact on the trajectories of the bodies over time. This "restricted" three body problem is made up of twelve equations, where there are four for each body. These equations can be seen below: \begin{align*} x_1'&=v_{1x}\\ v_{1x}'&=\frac{gm_2(x_2-x_1)}{((x_2-x_1)^2+(y_2-y_1)^2)^{3/2}}+\frac{gm_3(x_3-x_1)}{((x_3-x_1)^2+(y_3-y_1)^2)^{3/2}}\\ y_1'&=v_{1y}\\ v_{1y}&=\frac{gm_2(y_2-y_1)}{((x_2-x_1)^2+(y_2-y_1)^2)^{3/2}}+\frac{gm_3(y_3-y_1)}{((x_3-x_1)^2+(y_3-y_1)^2)^{3/2}}\\ x_2'&=v_{2x}\\ v_{2x}'&=\frac{gm_2(x_1-x_2)}{((x_2-x_1)^2+(y_2-y_1)^2)^{3/2}}+\frac{gm_3(x_3-x_2)}{((x_3-x_2)^2+(y_3-y_2)^2)^{3/2}}\\ y_2'&=v_{2y}\\ v_{2y}&=\frac{gm_2(y_1-y_2)}{((x_2-x_1)^2+(y_2-y_1)^2)^{3/2}}+\frac{gm_3(y_3-y_2)}{((x_3-x_2)^2+(y_3-y_2)^2)^{3/2}}\\ x_3'&=v_{3x}\\ v_{3x}'&=\frac{gm_2(x_1-x_3)}{((x_3-x_1)^2+(y_3-y_1)^2)^{3/2}}+\frac{gm_3(x_2-x_3)}{((x_3-x_1)^2+(y_3-y_1)^2)^{3/2}}\\ y_3'&=v_{3y}\\ v_{3y}&=\frac{gm_2(y_1-y_3)}{((x_3-x_1)^2+(y_3-y_1)^2)^{3/2}}+\frac{gm_3(y_2-y_3)}{((x_3-x_1)^2+(y_3-y_1)^2)^{3/2}}\\ \end{align*} Using this, we set \(m_1\)=0.3, and \(m_2=m_3=0.03\). The initial conditions were \((x_1,y_1)=(2,2)\),\((x_1',y_1')=(0.2,-0.2)\), \((x_2,y_2)=(0,0)\), \((x_2',y_2')=(0,0)\), \((x_3,y_3)=(-2,-2)\), and \((x_3',y_3')=(-0.2,0.2)\). The code for this can be seen in orbit_3body.m. The video below shows the animation for restricted three body problem.
The initial condition was then slightly changed for \(x_1'\) such that it was now equal to 0.20001. The resulting trajectories were compared, and the difference can be seen below.