Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 316 / 2 / ii

M_{1}

by the factor

1 - β_{1}

, so

\ddot{r} + 2 \hat{z} \times \dot{r} + \hat{z} \times (\hat{z} \times r) = - \frac{μ _{1} ( 1 - β _{1} )}{r _{1}^{3}} (r - r_{1}) - \frac{μ _{2}}{r _{2}^{3}} (r - r_{2}) .

U = \frac{1}{2} (x^{2} + y^{2}) + \frac{μ _{1} ( 1 - β _{1} )}{r _{1}} + \frac{μ _{2}}{r _{2}},

its Cartesian components are exactly

\overset{x}{¨} - 2 \overset{y}{˙} = U_{x}, \overset{y}{¨} + 2 \overset{x}{˙} = U_{y}, \overset{z}{¨} = U_{z} .

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