 Theory of the N-Body Problem
June 9, 1996
8
Studying the structure of the N-body problem shows that when bodies move under
one of the physical forces, the paths will be smooth and very predictable in the short
run.
(12:4)
This is the source of almost all the predictability that we see in everyday life:
objects fall down, rocks are hard, tides in the oceans, the seasons of the year, how
machines function, etc.
Studying the structure of the N-body problem also shows that in the long run
1
, it is
very hard to predict how things will turn out and that very small differences in initial con-
ditions can lead to wildly different results. This is the source of almost all of the chaos that
we see in everyday life: the weather, the roll of a die, the way things crack and break, etc.
It is this conflict between the predictability and the chaos, the glimmer of almost
knowing how a star pattern will turn out, but never being sure that lead me to develop the
XStar program to its current form.
1.2 Newtonian Physics
Newton laid out the formulas needed to solve the N-body problem for gravity
some 300 years ago. They are really fairly simple. The formulas are:
(16:762-85,17:78-83)
The position of the body.
Velocity is the rate of change of the position.
Acceleration is the rate of change of the velocity and
is also the second derivative of the position.
Force equals the mass times the acceleration.
The force of gravity between two bodies (of mass
m
1
and
m
2
) is equal to a constant
G
times the
product of the masses, divided by the square of the
distance
r
12
between the bodies. Technically, the
formula looks more like
where
is
the vector between the two bodies and
is the
length of the vector. That is, the force is projected
along the line connecting the two bodies.
1. While it is
possible
to give precise definitions of "short run" and "long run", it is not very useful to do so.
Just be aware that what is the "long run" for some problems might be the "short run" for others.
x
v
x
'
=
a
v
'
x
''
=
=
F
ma
=
F
Gm
1
m
2
r
12
2
=
F
Gm
1
m
2
r
12
2
r
12
r
12
=
r
12
r
12