Editing Celestial mechanics (section)

===Three-body problem===
{{main article | Three-body problem}}
After Newton, [[Joseph-Louis Lagrange]] attempted to solve the three-body problem in 1772, analyzed the stability of planetary orbits, and discovered the existence of the [[Lagrange point]]s. Lagrange also reformulated the principles of [[classical mechanics]], emphasizing energy more than force, and developing a [[Lagrangian mechanics|method]] to use a single polar coordinate equation to describe any orbit, even those that are parabolic and hyperbolic. This is useful for calculating the behaviour of planets and [[comet]]s and such (parabolic and hyperbolic orbits are [[conic section]] extensions of Kepler's [[elliptical orbit]]s). More recently, it has also become useful to calculate [[spacecraft]] [[trajectory|trajectories]].

[[Henri Poincaré]] published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). In them, he successfully applied the results of their research to the problem of the motion of three bodies and studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on). Poincaré showed that the three-body problem is not integrable. In other words, the general solution of the three-body problem can not be expressed in terms of [[algebra]]ic and [[transcendental functions]] through unambiguous coordinates and velocities of the bodies. His work in this area was the first major achievement in celestial mechanics since Isaac Newton.<ref>J. Stillwell, Mathematics and its history, [https://books.google.com/books?id=V7mxZqjs5yUC&pg=PA254 page 254]</ref>

These monographs include an idea of Poincaré, which later became the basis for mathematical "[[chaos theory]]" (see, in particular, the [[Poincaré recurrence theorem]]) and the general theory of [[dynamical system]]s. He introduced the important concept of [[Bifurcation theory|bifurcation points]] and proved the existence of equilibrium figures such as the non-ellipsoids, including ring-shaped and pear-shaped figures, and their stability. For this discovery, Poincaré received the Gold Medal of the Royal Astronomical Society (1900).<ref name="Gold Medal to Poincaré">{{cite journal |date=1900 |title=Address Delivered by the President, Professor G. H. Darwin, on presenting the Gold Medal of the Society to M. H. Poincaré |journal=Monthly Notices of the Royal Astronomical Society |volume=60 |issue=5 |pages=406–416 |doi=10.1093/mnras/60.5.406 |issn=0035-8711 |doi-access=free |author-last=Darwin |author-first=G.H.}}</ref>