Theory of the N-Body Problem

June 9, 1996

17

In order to calculate how the stars will move, we need to find the current accelera-

tion based on where a star is at. A little bit of simple algebra on Newton's formulas shows

that the acceleration for a given star due to one other star is

. For the N-body

problem, one copy of the right hand term is needed for each other star in the system. The

result is the vector equation:

(12:50)

Where
is the vector to the i

th

star.

While this is a vector equation, each dimension is handled in an identical manner

and the dimensions are related to each other only by the change to the overall distance

between the bodies. As was seen in Section
1.2.3, both dimensions could be looked at sep-

arately and both had similar functions. So, for the sake of simplicity, the vector nature of

the N-body problem will be ignored for the rest of the document.

It should be noted that just because the vector nature of the problem can, for the

most part, be ignored, the number of dimensions does have a significant impact on the

nature of the N-body problem. When there is only one dimension, stars must move along a

single line and therefore end up either colliding or going off to infinity. With two dimen-

sions, stable orbits can be created, and it is possible for unstable systems to last for an

indefinite amount of time. With three dimensions, the number of collisions is reduced even

further. After all, in order for stars to collide, they have to be close in all three dimensions

instead of just two.

The above formula is a messy enough as is, but as shown, it doesn't even taken

into account the fact that the position, velocity and acceleration all vary with time. So, this

equation will be abbreviated as:

Where

*f()*

is the complicated force function

This equation is known as a "differential equation" because it relates the second

derivative of

, to the position
at time

*t*

. More specifically, this is known as an Ordinary

Differential Equation, or ODE

1

. Sometimes there are ways of solving ODEs that come up

with exact formulas, but for the N-body problem with more than two bodies, no one has

found a method yet. When a problem can't be solved through the standard methods of dif-

ferential equations, you generally have to resort to numerical analysis

2

.

(12:49,2:233,1:289)

1. There are also Partial Differential Equations (PDEs), but they have no bearing on the N-body problem.

2. Numerical analysis is the study of finding approximate solutions to equations through calculations as

opposed to symbol manipulation like you would do with algebra.

*x*

''

*G**m*

2

*r*

12

2
3

*r*

12

=

*x*

''

*G**m*

*i*

*r*

*i*

2
3

*r*

*i*

*i*

1

=

*n*

=

*r*

*i*

*x*

''

*t*

(
)

*f*
*x*
*t*

(
)

(

)

=

*x*

''

*x*

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