Sir Isaac Newton and modern celestial mechanics
Authors: Szebehely Victor 
Published: June 2020

This address contains a concise review of the most recent and modern accomplishments of the classical science of Celestial Mechanics to which Sir Isaac Newton contributed its foundations. Celestial Mechanics is the scientific discipline dedicated to the study of the dynamical behavior of natural and artificial celestial bodies. It was originally considered part of astronomy, (" gravitational astronomy " as it was often called) but its applications to geodynamics, space exploration, astrodynamics, orbital mechanics, stellar dynamics, and artificial satellite dynamics, suggest a necessarily more general field.
Almost all of the pre-Newtonian results are obtainable from Newton's laws of motion and his universal law of gravitation. What is known today as the direct problem of Celestial Mechanics might be formulated in terms of systems of ordinary differential equations. Finding the solutions (representing the time-dependent motions of the participating bodies) of these differential equations for given sets of initial conditions is the aim of Celestial Mechanics. The inverse problem is the establishment of the force fields which result in observed orbits.
The accomplishments along both of these lines are most impressive since Newton, and are due to his basic contributions, aswell as to advances along analytical, computational and observational lines.
What is often called today the " Newtonian deterministic " approach (described above as the direct problem) was reemphasized by Laplace and was generally accepted in Celestial Mechanics until recently. Note that Newton him¬ self questioned the deterministic approach which was completely adopted by Leibnitz creating the second conflict between them, besides the claim of dis¬ covery of calculus. Other fields of science recognized limitations of predic¬ tability and accepted non-deterministic approaches long before the last Citadel of Determinism, Celestial Mechanics was penetrated by representatives of the Club of Statistical Craze (as Einstein called it).
Today we have our modern non-deterministic Celestial Mechanics, soundly based on Newtonian principles, but at the same time, recognizing the necessity of possible modification of force laws (relativity effects, higher order gravitation terms, drags, etc.), understanding the difficulties of analytical approaches (Poin-caré's non-in tegrability), accepting the limitations of numerical techniques (finite number of digits, error accumulations), realizing the statistical nature of the initial conditions (given with uncertainties) and introducing the primary importance of numerical, structural and dynamical stability. One of the significant consequences of these modern recognitions of realism in Celestial Mechanics is the appearance of a set of trajectories representing the solution of a problem as opposed to a single orbit. As time or a corresponding independent variable changes, the members of the ensemble of trajectories will deviate depending on the stability of the system and might show chaotic behavior. At this point the system " will not remember " its initial conditions and reliable predictions become the figments of our imagination.
The basic formulation of Celestial Mechanics is still Newtonian. The interpretation of solutions, the problems of primary interest, the analytical and numerical approaches, the applications and the observational techniques have been enlarged.