Celestial mechanics

Prior to Kepler, there was little connection between exact, quantitative prediction of planetary positions, using geometrical or numerical techniques, and contemporary discussions of the physical causes of the planets' motion.

Kepler's elliptical model greatly improved the accuracy of predictions of planetary motion, years before Newton developed his law of gravitation in 1686.

Isaac Newton is credited with introducing the idea that the motion of objects in the heavens, such as planets, the Sun, and the Moon, and the motion of objects on the ground, like cannon balls and falling apples, could be described by the same set of physical laws.

Lagrange also reformulated the principles of classical mechanics, emphasizing energy more than force, and developing a method to use a single polar coordinate equation to describe any orbit, even those that are parabolic and hyperbolic.

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).

In other words, the general solution of the three-body problem can not be expressed in terms of algebraic and transcendental functions through unambiguous coordinates and velocities of the bodies.

[2] Simon Newcomb was a Canadian-American astronomer who revised Peter Andreas Hansen's table of lunar positions.

By the time he attended a standardisation conference in Paris, France, in May 1886, the international consensus was that all ephemerides should be based on Newcomb's calculations.

Albert Einstein explained the anomalous precession of Mercury's perihelion in his 1916 paper The Foundation of the General Theory of Relativity.

General relativity led astronomers to recognize that Newtonian mechanics did not provide the highest accuracy.

Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly.

The earliest use of modern perturbation theory was to deal with the otherwise unsolvable mathematical problems of celestial mechanics: Newton's solution for the orbit of the Moon, which moves noticeably differently from a simple Keplerian ellipse because of the competing gravitation of the Earth and the Sun.

Perturbation methods start with a simplified form of the original problem, which is carefully chosen to be exactly solvable.

"[6] This general procedure – starting with a simplified problem and gradually adding corrections that make the starting point of the corrected problem closer to the real situation – is a widely used mathematical tool in advanced sciences and engineering.