the two-body (Kepler) problem
/ KEP-ler /
Look up at a planet circling the Sun, or the Moon circling the Earth: two bodies, pulling on each other by gravity, dancing in space. The two-body problem asks the most fundamental question in celestial mechanics — given two masses and the law of gravity, exactly what path do they trace? Its answer, worked out from Newton's laws, is one of the supreme triumphs of differential equations: it explained the planets, vindicated Kepler's observed laws, and launched mathematical physics.
Two masses attract along the line between them with a force that weakens as the square of their separation r: gravity gives r'' proportional to -1/r^2 directed inward (the inverse-square law). Two simplifications crack it open. First, the centre of mass drifts steadily and can be ignored, so you only track the RELATIVE vector between the bodies — turning two bodies into one effective body orbiting a fixed centre. Second, gravity is a central force, so angular momentum is conserved, which pins the motion to a plane and gives Kepler's law of equal areas in equal times for free.
Solve the resulting equations and out come Kepler's three laws as theorems, not guesses. The orbit is always a conic section — an ellipse for a bound planet (with the Sun at one focus), a parabola or hyperbola for an unbound comet that swings by once and leaves. The orbital period squared is proportional to the semi-major axis cubed. That gravity ALONE, an inverse-square pull, forces orbits to be exact ellipses is a near-miraculous fact, special to the 1/r^2 law and to no other power.
The honesty: the clean closed-form solution exists ONLY for two bodies. Add a third — Sun, Earth, and Moon together — and the three-body problem has no general elementary solution and can be chaotic. So the tidy ellipses of the textbook are an idealization: real planets perturb one another, orbits slowly precess, and the solar system's exact long-term future is, strictly, beyond closed-form reach.
The Earth orbits the Sun on an ellipse with the Sun at one focus, sweeping equal areas in equal times — so it moves fastest at its closest approach in January and slowest at its farthest point in July. Both facts drop straight out of the inverse-square two-body equations, with no curve-fitting.
Inverse-square gravity forces a closed elliptical orbit with the Sun at a focus.
The exact closed-form ellipse is a special prize of the TWO-body, inverse-square problem. Three or more gravitating bodies have no general elementary solution and can behave chaotically, so real planetary orbits only approximate the clean Keplerian ellipse.