Newton's theorem of revolving orbits

Newton's theorem shows how changing a central force by an inverse-cube term makes a body trace the same orbit with its apsides revolving at a chosen rate.

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Central force: A force directed along the line joining a body to a fixed or moving center, with magnitude depending on position or distance. The theorem applies to motion under forces aimed at a center.
Apsis: A point on an orbit where the distance to its central body reaches a local maximum or minimum. The apsides are the endpoints defining the line that revolves.
Isaac Newton: English mathematician and physicist whose 1687 work formulated classical mechanics and universal gravitation. Newton stated the revolving-orbit result in his mechanics.
Mercury's perihelion precession: The gradual rotation of Mercury's closest-approach direction to the Sun, including a relativistic contribution unexplained by Newtonian planetary perturbations. It is a prominent observed case of apsidal motion, though not explained by Newton's theorem alone.
Kepler problem: The problem of determining motion under an inverse-square central force. It is the baseline dynamical system whose orbit shape the theorem modifies.
Apsidal precession: The rotation of an orbit's line of apsides, the line joining its nearest and farthest points from the force center. This is the orbital change Newton's force modification prescribes.
Orbit: The path of an object moving under the influence of gravity or another force. The theorem begins with a specified orbit to be preserved.
Philosophiæ Naturalis Principia Mathematica: Isaac Newton's 1687 treatise presenting the laws of motion and universal gravitation. The theorem appears in Newton's systematic treatment of orbital dynamics.
General relativity: Albert Einstein's theory of gravity, describing gravitation as the curvature of spacetime. Its correction to Mercury's orbit is often compared with Newtonian inverse-cube force models.
Harmonic oscillator: A system in which restoring force is proportional to displacement from equilibrium. Its central-force version, like the Kepler force, produces closed bound orbits.