Theory of the N-Body Problem
June 9, 1996
18
The N-body problem is also known as an "Initial Value Problem" because, typi-
cally, the locations and velocities of the objects are known at a given time and the problem
is to determine the state of the star system for some time in the future. It can be shown that
the N-body problem satisfies the Lipschitz condition, and therefore giving any particular
initial situation, there can only be one possible outcome. While this might be expected for
the N-body problem, it is not always true for all types of ODEs.
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One way of working with ODEs is to integrate out the derivatives, as follows. In
the case of the N-body problem, the integration has to be done twice, once to get the
velocity, once more to get the position.
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The integration of
can then be performed by first breaking time down into a
series of small, discrete steps, a process that is known as "discretization" of the problem.
Then approximate solutions for each step in the series can be calculated and added
together. This, of course, leads to small "discretization errors" which accumulate and can
lead the future state of the bodies away from the one true future state.
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The goal is to make these errors be as small as possible and one way of reducing
the discretization error is to reduce the size of each step. There are two problems with this
solution though. First, it takes longer to come up with the final solution. Secondly, there is
another source of errors, known as round off errors, which are inherent in (almost) all
floating point calculations on computers. The smaller the step size, the more calculations
will have to be done and the larger the accumulated rounding errors will be. So, there is a
limit to how small the step size can be made without starting to increase the size of the
total; error again.
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A second problem with integrating equation (EQ 1) is that it is hard to sample the
value of
during the time period that we want to integrate over because the circular
dependency between
and
. This makes the standard methods of numeric inte-
grate difficult to use. There are two basic approaches that can be used to work around this
problem. The first method is to use an extrapolating or open ended method of integration
where only the values of
from the past and present are used to integrate each step.
The second method is to consider the integral in equation (EQ 1) as an unknown function.
Even though we don't know what this function is, at least the first few derivatives are easy
to find, and it is theoretically possible to find all derivatives. This property means that the
Taylor series can be used to approximate the function.
x
''
t
( )
f x t
( )
(
)
=
t
2
2
d
d x t
( )
f x t
( )
(
)
=
d
2
x t
( )
f x t
( )
(
)
t
2
d
=
x t
( )
f x t
( )
(
)
t t
d
d
=
(EQ 1)
f x t
( )
(
)
f x t
( )
(
)
x t
( )
f x t
( )
(
)
f x t
( )
(
)