Central force motion and orbits

A central force is any force whose direction always points exactly toward or away from one fixed point — gravity from a star, the electric force between two point charges, a mass on an ideal spring anchored at the origin. The defining geometric fact about such a force is what makes it special: torque is force times the perpendicular lever arm, and a force pointing directly along the line to the pivot has zero perpendicular component about that pivot, whatever the force's strength or how it depends on distance. Zero torque about the centre means angular momentum about that centre is exactly conserved, throughout the entire motion — not approximately, and not only for gravity's particular 1/r² form, but for any central force whatsoever.

That conservation law is what gives orbital motion its most useful structural simplification. Angular momentum L = mvr sin θ being fixed links speed and distance together at every point of the path, which is precisely Kepler's second law in more general clothing: closer to the centre, the object must move faster to keep L constant, and farther away it moves slower, sweeping equal areas in equal times regardless of the orbit's exact shape. Rather than tracking a full two-dimensional trajectory, the conserved angular momentum lets the problem be reduced to a single equation in one variable, the radial distance from the centre — the angular part of the motion is already determined once L is known.

This is why every central force — not just gravity — produces orbits confined to a plane (since angular momentum has a fixed direction as well as magnitude) that sweep out area at a constant rate. What differs between different central forces is only the shape the orbit ultimately takes: gravity's 1/r² law produces the closed ellipses of Kepler's first law, while other central forces produce different, sometimes non-closed paths — the conservation law is universal to the whole family, but the resulting geometry depends on which particular force is doing the pulling.