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The 3-body problem in six equations $$\newcommand{\norm}[1]{\lVert#1\rVert}$$

Tags. differential, gravitation, mechanics, newtonian, three-body

Let $$m_0$$, $$m_1$$ and $$m_2$$ be point-like masses, $$X^i = (X^1,X^2,X^3)$$ the $$i$$ components of the (cartesian) position vector $$\boldsymbol{X}$$ of the centre of mass $$cm(1,2)$$ of particles $$m_1$$ and $$m_2$$ relative to $$m_0$$, and $$x^i = (x^1,x^2,x^3)$$ the $$i$$ components of the position vector $$\boldsymbol{x}$$ of $$m_2$$ relative to $$m_1$$, as depicted in the figure.

These coordinates are sufficient to determine the 6 equations that describe the motion of the gravitational 3-body system: \begin{align*} \ddot X^i &= - G \frac{M {m_1}^3}{m_1 + m_2} \frac{m_1 X^i - \mu x^i}{{\norm{m_1 \pmb X - \mu \pmb x}}^3} -G \frac{M {m_2}^3}{m_1 + m_2} \frac{m_2 X^i + \mu x^i}{{\norm{m_2 \pmb X + \mu \pmb x}}^3}\\[2ex] \ddot x^i &= \hspace{0.5em} G m_0 {m_1}^2 \frac{m_1 X^i - \mu x^i}{{\norm{m_1 \pmb X - \mu \pmb x}}^3} - G m_0 {m_2}^2 \frac{m_2 X^i + \mu x^i}{{\norm{m_2 \pmb X + \mu \pmb x}}^3} - G (m_1 + m_2) \frac{x^i}{\norm{\pmb x}^3} \end{align*} where \begin{equation*} \mu = \frac{m_1 m_2}{m_1 + m_2} \end{equation*} and $$M = \sum_i m_i$$ is the total mass of the system.

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