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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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