I am trying to understand three body problem in Newtonian space. I want to make formulation of differential equations for known initial conditions for the case with:
- Identical three masses
- Periodic solution
- Zero Angular momentum
The equations in vector form are:
$$\frac{d^2 \overset{\rightharpoonup }{r}_1}{\text{dt}^2}=\frac{k^2 m_2 \left(\overset{\rightharpoonup }{r}_2-\overset{\rightharpoonup }{r}_1\right)}{r_{12}^3}+\frac{k^2 m_3 \left(\overset{\rightharpoonup }{r}_3-\overset{\rightharpoonup }{r}_1\right)}{r_{13}^3}$$
$$\frac{d^2 \overset{\rightharpoonup }{r}_2}{\text{dt}^2}=\frac{k^2 m_1 \left(\overset{\rightharpoonup }{r}_1-\overset{\rightharpoonup }{r}_2\right)}{r_{12}^3}+\frac{k^2 m_3 \left(\overset{\rightharpoonup }{r}_3-\overset{\rightharpoonup }{r}_2\right)}{r_{23}^3}$$
$$\frac{d^2 \overset{\rightharpoonup }{r}_3}{\text{dt}^2}=\frac{k^2 m_1 \left(\overset{\rightharpoonup }{r}_1-\overset{\rightharpoonup }{r}_3\right)}{r_{13}^3}+\frac{k^2 m_2 \left(\overset{\rightharpoonup }{r}_2-\overset{\rightharpoonup }{r}_3\right)}{r_{23}^3}$$
where $k$ denotes the gravity constant.
I want to formulate (not solve) this problem properly.
I've found this reference can help understand zero angular momentum. I don't understand it well.
