N-body problem

In physics, the $n$ -body problem is the problem of predicting the individual motions of a group of celestial objects interacting with each other gravitationally.^[1] Solving this problem has been motivated by the desire to understand the motions of the Sun, Moon, planets, and visible stars. In the 20th century, understanding the dynamics of globular cluster star systems became an important $n$ -body problem.^[2] The $n$ -body problem in general relativity is considerably more difficult to solve due to additional factors like time and space distortions.

The classical physical problem can be informally stated as the following:

Given the quasi-steady orbital properties (instantaneous position, velocity and time)^[3] of a group of celestial bodies, predict their interactive forces; and consequently, predict their true orbital motions for all future times.^[4]

The two-body problem has been completely solved and is discussed below, as well as the famous restricted three-body problem.^[5]

^ Leimanis and Minorsky: Our interest is with Leimanis, who first discusses some history about the $n$ -body problem, especially Ms. Kovalevskaya's 1868–1888 twenty-year complex-variables approach, failure; Section 1: "The Dynamics of Rigid Bodies and Mathematical Exterior Ballistics" (Chapter 1, "The motion of a rigid body about a fixed point (Euler and Poisson equations)"; Chapter 2, "Mathematical Exterior Ballistics"), good precursor background to the $n$ -body problem; Section 2: "Celestial Mechanics" (Chapter 1, "The Uniformization of the Three-body Problem (Restricted Three-body Problem)"; Chapter 2, "Capture in the Three-Body Problem"; Chapter 3, "Generalized $n$ -body Problem").
^ See references cited for Heggie and Hut.
^ Quasi-steady loads are the instantaneous inertial loads generated by instantaneous angular velocities and accelerations, as well as translational accelerations (9 variables). It is as though one took a photograph, which also recorded the instantaneous position and properties of motion. In contrast, under a steady-state condition, a system's state is invariant to time; otherwise, the first derivatives and all higher derivatives are zero.
^ R. M. Rosenberg states the $n$ -body problem similarly (see References): "Each particle in a system of a finite number of particles is subjected to a Newtonian gravitational attraction from all the other particles, and to no other forces. If the initial state of the system is given, how will the particles move?" Rosenberg failed to realize, like everyone else, that it is necessary to determine the forces first before the motions can be determined.
^ A general, classical solution in terms of first integrals is known to be impossible. An exact theoretical solution for arbitrary $n$ can be approximated via Taylor series, but in practice such an infinite series must be truncated, giving at best only an approximate solution; and an approach now obsolete. In addition, the $n$ -body problem may be solved using numerical integration, but these, too, are approximate solutions; and again obsolete. See Sverre J. Aarseth's book Gravitational $n$ -Body Simulations listed in the References.

[Leimanis_and_Minorsky-1] Leimanis and Minorsky: Our interest is with Leimanis, who first discusses some history about the $n$ -body problem, especially Ms. Kovalevskaya's 1868–1888 twenty-year complex-variables approach, failure; Section 1: "The Dynamics of Rigid Bodies and Mathematical Exterior Ballistics" (Chapter 1, "The motion of a rigid body about a fixed point (Euler and Poisson equations)"; Chapter 2, "Mathematical Exterior Ballistics"), good precursor background to the $n$ -body problem; Section 2: "Celestial Mechanics" (Chapter 1, "The Uniformization of the Three-body Problem (Restricted Three-body Problem)"; Chapter 2, "Capture in the Three-Body Problem"; Chapter 3, "Generalized $n$ -body Problem").

[Heggie_and_Hut-2] See references cited for Heggie and Hut.

[3] Quasi-steady loads are the instantaneous inertial loads generated by instantaneous angular velocities and accelerations, as well as translational accelerations (9 variables). It is as though one took a photograph, which also recorded the instantaneous position and properties of motion. In contrast, under a steady-state condition, a system's state is invariant to time; otherwise, the first derivatives and all higher derivatives are zero.

[4] R. M. Rosenberg states the $n$ -body problem similarly (see References): "Each particle in a system of a finite number of particles is subjected to a Newtonian gravitational attraction from all the other particles, and to no other forces. If the initial state of the system is given, how will the particles move?" Rosenberg failed to realize, like everyone else, that it is necessary to determine the forces first before the motions can be determined.

[5] A general, classical solution in terms of first integrals is known to be impossible. An exact theoretical solution for arbitrary $n$ can be approximated via Taylor series, but in practice such an infinite series must be truncated, giving at best only an approximate solution; and an approach now obsolete. In addition, the $n$ -body problem may be solved using numerical integration, but these, too, are approximate solutions; and again obsolete. See Sverre J. Aarseth's book Gravitational $n$ -Body Simulations listed in the References.

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