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Two-Body Problem Formulation

The foundational mathematical model describing the motion of two masses interacting solely via Newtonian gravity. It simplifies the N-body problem and yields closed-form solutions for the orbit's geometry and dynamics.
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The statement of the theorem

Consider two masses m1m_1 and m2m_2 separated by a relative position vector r=r2−r1\mathbf{r} = \mathbf{r}_2 - \mathbf{r}_1. The equation of motion for the relative vector r\mathbf{r} is governed by the differential equation:\nd2rdt2=−μr3r\frac{d^2\mathbf{r}}{dt^2} = -\frac{\mu}{r^3} \mathbf{r} \nwhere μ=G(m1+m2)\mu = G(m_1 + m_2) is the gravitational parameter, and r=∣∣r∣∣r = ||\mathbf{r}||. This equation yields closed-form solutions describing conic sections.