N-body problem

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The N-body problem is a physical problem of classical mechanics , which aims to set up equations of motion for each individual mass point. The N-body problem is mostly used by astronomers to simulate the movement of planets , stars, satellites, etc. This is why the classic N-body problem is still used today in astronomy for simple calculations. Simulations are called N-body simulation .

The N-body problem of general relativity is much more difficult to solve than that of classical mechanics , which is why the classical model is still used for many simulations.

The most important special case of the N-body problem is the two-body problem ( ), which was already solved in the 17th century and has since been used to calculate the path of two bodies.

General formulation

Given are point masses ( ) that move in three-dimensional space under their mutual gravitational influence. The position of the -th mass point is given by the position vector .

According to Newton's second law is

equal to the sum of the forces acting on the particle , in this case the gravitational forces of all other particles on the -th.

The gravitational interaction between the -th and the -th particle is given by Newton's law of gravitation

With this we can write the equations of motion as follows:

Whereby the potential is given by

With the canonical impulse

and the canonical coordinates can be derived from Hamilton's equations of motion :

write, taking the Hamilton function through

is defined. is the kinetic energy of the system:

From the Hamilton equations we see that the body problem can be described by a system of explicit ordinary differential equations of first order .

Special cases

The two-body problem

The two-body problem is particularly important in astronomy, as it can describe the orbits of two planets etc. with great accuracy .

Movement of the center of gravity

To solve the two-body problem, we first set up Newton's equations of motion for the two particles:

By adding the two equations of motion we get:

After introducing centroid coordinates we can solve the two-body problem

The center of gravity of the two-body system moves in a straight line uniformly.

Movement of the mass points

In addition to determining the movement of the center of gravity, the determination of the movement of the individual mass points is sometimes referred to as a two-body problem. This problem is mathematically more complex, which is why only the solution is outlined here.

From the differential equation

we get . This reduces the two-body problem to the determination of and from .

If these are known, the movement of the mass points can be determined

and

determine.

The three-body problem

Newton introduced the first definitions and theorems of the three-body problem as early as 1687 in his “ Principia ”. Since then, numerous special solutions have been found. The first of these solutions was found in 1767 by Leonhard Euler . Only five years later (1772) the physicist Joseph-Louis Lagrange found another solution for objects that form an equilateral triangle. In this solution, the Lagrange points were also introduced for the first time .

There are no closed analytical solutions for the general three-body problem, since the movement of the bodies forms a chaotic system for most of the initial values ​​and therefore numerical solutions have to be used. In general, the movement of the body is also non-periodic.

General solution

In the physical literature, the N -body problem is sometimes referred to as "unsolvable". However, this formulation should be used with caution, as “unsolvable” is not clearly defined. For the N -body problem with , Henri Poincaré showed that there can be no closed solution (such as the elliptical orbits of the bound Kepler problem).

The N-particle problem with the Taylor series

The N-particle problem can u. a. solve by introducing a Taylor expansion.

We define our system of differential equations as follows:

Since and are known as initial values, we also know . By differentiating again, we then also know the higher derivatives, which means that the Taylor series is known as a whole. However, it remains to show what the radius of convergence of this series is and especially how it behaves in view of the poles (the right side of the equation of motion diverges when two mass points come arbitrarily close). The Chinese physicist Wang Qiu-Dong solved this question in 1991 by transforming the time coordinate so that singularities only occur at. The solution found is of no practical importance, however, since the series found in this way converge extremely slowly. New theoretical statements, for example about the stability of the N -body problem, have not yet resulted from this solution.

simulation

In addition to the analytical solution of N-body problems, there are also numerical methods. With these, many analytically difficult to solve problems can be solved quite easily.

Individual evidence

  1. Basic course Theoretical Physics 1 . In: Springer textbook . 2006, doi : 10.1007 / 978-3-540-34833-7 .
  2. V. Analytical Mechanics . In: Theoretical Physics / Mechanics . DE GRUYTER, Berlin, Boston, ISBN 978-3-11-083533-5 , pp. 102-123 , doi : 10.1515 / 9783110835335.102 .
  3. Timothy Gowers, June Barrow-Green, Imre Leader: V.33 The Three-Body Problem . In: The Princeton Companion to Mathematics . Princeton University Press, Princeton 2010, ISBN 978-1-4008-3039-8 , pp. 726-728 , doi : 10.1515 / 9781400830398.726 (English).
  4. a b Christoph Pöppe: The solution to the n-body problem . In: Spectrum of Science . No. 1 , 1997, p. 24 ( Spektrum.de ).
  5. ^ Wang Qiu-Dong: The Global Solution of the n-Body Problem . In: Celestial Mechanics and Dynamical Astronomy . tape 50 , 1991, pp. 73-88 .