Celestial mechanics is the branch of astronomy and applied mathematics concerned with the motions of celestial bodies under the influence of gravitational and, where relevant, non-gravitational forces. Its scope encompasses the orbits of planets, moons, comets, asteroids, and artificial satellites, as well as the long-term dynamical evolution of planetary systems, tidal interactions, and the behavior of multi-body gravitational systems. The field draws on classical Newtonian mechanics, general relativity, and numerical methods to predict and explain the positions and trajectories of bodies across timescales ranging from hours to billions of years.
The mathematical foundations of celestial mechanics were laid in the 17th and 18th centuries, principally through the work of Johannes Kepler, Isaac Newton, and later Leonhard Euler, Joseph-Louis Lagrange, and Pierre-Simon Laplace. Newton's law of universal gravitation and his formulation of the three laws of motion provided the first unified quantitative framework for planetary motion. The subsequent development of perturbation theory allowed astronomers to account for the gravitational influence of multiple bodies on one another, enabling highly accurate predictions of planetary positions. For a detailed account, see Celestial Mechanics - History.
Modern celestial mechanics is divided into several sub-disciplines. Analytical celestial mechanics seeks closed-form or series solutions to equations of motion, while numerical celestial mechanics relies on computational integration of those equations over time. Astrodynamics applies the principles of celestial mechanics to the trajectories of spacecraft. Dynamical astronomy addresses the large-scale structure and evolution of gravitating systems, including the stability of the solar system over geological and cosmological timescales.
The two-body problem - the motion of two point masses interacting only with each other - admits an exact solution in Newtonian gravity: the bodies follow conic sections (ellipses, parabolas, or hyperbolas) as described by Kepler's laws. The three-body problem, by contrast, has no general closed-form solution; Henri Poincare's late-19th-century work demonstrated that its solutions can be chaotic, a result foundational to modern dynamical systems theory. N-body problems with four or more bodies are treated primarily through numerical integration and statistical methods.
General relativity modifies the predictions of Newtonian gravity in regimes of strong gravitational fields or high velocities. The perihelion precession of Mercury, unexplained under Newtonian mechanics, was one of the first empirical confirmations of Einstein's general theory. For most solar system applications, Newtonian mechanics with relativistic correction terms provides sufficient accuracy; fully relativistic treatment is required for phenomena such as binary pulsar timing, gravitational wave emission from compact binary systems navigated using celestial mechanics methods, and spacecraft navigation near massive bodies.
The core mathematical framework of Newtonian celestial mechanics is regarded as settled science within the physical sciences community. The applicability of general relativity to strong-field and high-precision contexts is well-established. See Celestial Mechanics - Physics Consensus for detail on the scope and limits of that consensus.