Theory of the N-Body Problem

June 9, 1996

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Newton's law of gravity says that there will be a force exerted by the two bodies

on each other. The force will be along the line that connects them and the forces must be

equal and opposite to each other. These forces will cause the bodies to accelerate toward

each other and the magnitude of that acceleration depends only on the distance and mass

of the other body. The acceleration will cause the velocities of the bodies to change and

cause the bodies to no longer move in a straight line. (See FIGURE 3.)

When there are only two bodies in a system, they will always move on paths that

follow one of the conic sections

1

. If the bodies are moving slowly enough that they orbit

each other, they will move around their common center of mass.

For a slightly more complicated example, let's look at the case of four bodies as in

FIGURE 4. Each body exerts a force on all other bodies. The total force, and therefore the

total acceleration is simply the sum of all three forces. The paths that the bodies will take

when there are more that two bodies can be very complicated.

In the following example, we will look at a simple two body system in some detail

One of the bodies will be so much more massive than the other body that it is effectively

immobile. This is similar to the case of a comet and the sun, or the earth and a satellite.

1. The conics sections are the shapes made when you cut a cone with a plane. Depending on the angle of the

plane, you can get a point, a single line, two intersecting lines, a circle, an ellipse, a parabola or a hyperbola.

**FIGURE 3. Paths caused by the gravitational force**

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2

**FIGURE 4. Forces exerted by several bodies**

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2

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