Theory of the N-Body Problem
June 9, 1996
17
1.3 Development of Methods to Solve the N-body Problem
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
''
Gm
2
r
12
2 3
r
12
=
x
''
Gm
i
r
i
2 3
r
i
i
1
=
n
=
r
i
x
''
t
( )
f x t
( )
(
)
=
x
''
x
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