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The general statement for the three body problem is as follows. At an instant in time, for vector positions <math>x_i</math> and masses <math>m_i</math>, three coupled second-order differential equations exist:<ref>[http://www.physics.buffalo.edu/phy410-505/2011/topic2/app2/ The Gravitational Three Body Problem]</ref><ref>[http://www.phys.lsu.edu/faculty/gonzalez/Teaching/Phys7221/ThreeBodyProblem.pdf The Three-Body Problem]</ref>
: <math>\ddot{x}_1 = -G m_2 \frac{x_1 - x_2}{|x_1 - x_2|^3} - G m_3 \frac{x_1 - x_3}{|x_1 - x_3|^3}</math>
: <math>\ddot{x}_2 = -G m_3 \frac{x_2 - x_3}{|x_2 - x_3|^3} - G m_1 \frac{x_2 - x_1}{|x_2 - x_1|^3}</math>
: <math>\ddot{x}_3 = -G m_1 \frac{x_3 - x_1}{|x_3 - x_1|^3} - G m_2 \frac{x_3 - x_2}{|x_3 - x_2|^3}</math>
A complete solution for a particular three-body problem provides the positions for all three particles for all time given three initial positions and initial velocities.
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