ZweikörperProblem

The ZweiKörperProblem refers to predicting the motion of two point masses interacting through Newtonian gravity. It is a foundational problem in celestial mechanics, illustrating how gravitational forces determine orbital motion and providing exact solutions for many practical situations, such as planets around the Sun or binary stars.

In the Newtonian framework, the problem can be reduced by moving to the center-of-mass frame and describing the relative coordinate r = r2 − r1. The two bodies are equivalent to a single particle of reduced mass μ = m1 m2/(m1 + m2) moving under a central potential V(r) = − G m1 m2 / r. The resulting equation of motion for the relative coordinate is d^2 r/dt^2 = − G M r / r^3, with M = m1 + m2. The angular momentum h = |r × v| is conserved, as is the total energy E = 1/2 μ v^2 − G m1 m2 / r.

The trajectory lies in a plane and can be written in polar form as r(θ) = p / (1 + e cos θ). Here p = h^2 /(G M) is the semilatus rectum, and e is the eccentricity given by e^2 = 1 + 2 E h^2 /(G M)^2. Negative E yields elliptical orbits (including circular when e = 0), E = 0 parabolic, and E > 0 hyperbolic trajectories.

Applications include modeling planetary and binary-star orbits. The problem also serves as a baseline for perturbations and relativistic effects, which introduce phenomena such as periapsis precession and deviations from purely Newtonian behavior.