Theory of the N-Body Problem

June 9, 1996

9

Using the above equations, it can be proven that several properties of a star system

can not be changed during the lifetime of the system. The proofs are fairly involved, but

the results are well known. Collectively, these properties are known as the "constants of

motion".

(12:49-56)

The constants of motion are:

*****

The total energy of the system must be conserved. So, if the kinetic energy of

the system increases, the potential energy must decrease.

*****

Matter can be neither created nor destroyed.

*****

The total (linear) momentum of the system must be conserved.

*****

The total angular momentum of the system must be conserved.

*****

The center of mass of the system, if it moves at all, must move in a straight

line and with a constant speed.

Knowing that these items must remain constant can be used to help determine if

the results from a "solution" to the N-body problem is correct and, if not, the size of the

error. It will be shown later that numerical solution can not, in general, be exactly correct,

so determining the type and amount of errors is an important part of creating a good

method for solving the N-body problem.

While these formulas are not very complicated, it can be hard to get a good feel for

how the formulas respond with out working with them a fair amount, so looking at a few

examples at this time is warranted.

As a first example, let's look at the case of just two bodies in space as shown in

FIGURE 3. Each body will have a position in space, a mass and a velocity, which are inde-

pendent of all other bodies. If there is no force applied to a body, it will continue along on

a straight line in the direction of the velocity vector. How quickly the body would move

depends on the size of the velocity vector. In this example, Body 1 is moving up and to the

left, Body 2 is moving down and to the left. Body 2 is also moving quicker than Body 1.

A body with a certain mass

Gravitational force

Velocity of the body

**FIGURE 2. The components of a two body system**

1

2

This document is best viewed as n-body.pdf because the translation to html was buggy.