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.
1.2.2 An Example of Newtonian Physics With Four Bodies
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.
1.2.3 A Detailed Example of Newtonian Physics With Two Bodies
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
3