Theory of the N-Body Problem

June 9, 1996

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The N-body problem has been studied since ancient times, although people didn't

realize that that was what they were doing. Any time there are several bodies (planets,

stars, apples, electrons, etc.) that move under the force of one of the physical laws, you

have an example of the N-body problem. When the ancient Greeks studied the movement

of the planets, the changes of the seasons or how gravity works here on Earth, they were

actually struggling to find solutions to the N-body problem. Significant headway on the

problem did not occur until Copernicus and Kepler tackled the problem in the mid 1500's

and it wasn't until Isaac Newton released his work,

*Principia*

in 1687 that a solution to the

special case of

was found. In this book, Newton laid out his laws of gravity and the

foundation of what we now know as classical physics.

(3:148,12:1)

The study of the N-body problem has had a significant impact on history. The orig-

inal theories of the movement of the stars lead to many religious and philosophical beliefs,

some of which are still around today, for example, in the form of astrology. The work that

Newton did on the N-body problem lead directly to the development of Integral and Dif-

ferential Calculus. His development of physics has lead directly to modern engineer-

ing.

(3:148)

One of the major justifications for building the first electronic computer systems

was to solve the N-body problem for artillery shell trajectories.

(9:23)

Modern physics has found that there are only four fundamental physical forces,

namely: gravity, electro-magnetic, strong nuclear, and weak nuclear

1

. These forces all

have a few things in common: they can be expressed in very simple formulas, they all are

proportional to some property of an object (mass, electrical charge, etc.), and they all get

weaker the further apart the objects are from each other

(7:30-33)

. How particles move under

these forces are all solved with similar N-body programs. Gravity is the simplest of these

forces, and the only one that is implemented by XStar and, so, it is the only one that will

be directly discussed. Besides the modifications needed to handle the other forces, there

are also specialized versions of the N-body problem that take Einstein's relativity into

account, and ones that take into account quantum mechanics.

When most physics text books present these forces, they show the simple formu-

las, and they use these formulas to solve many simple cases. Rarely do they try and

present cases where more than two objects are creating the forces and for good reason. As

soon as there are three objects that move under one of the four fundamental forces, the N-

body problem becomes very hard to solve. It is the N-body problem that links the four

physical forces to the complex results that we see as the universe.

(12:49)

1. Only gravity and the electro-magnetic forces are observed directly in everyday life. The strong force is

responsible for the different elements that atoms come in and for nuclear power. The weak nuclear force

causes certain types of nuclear decay.

*n*

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