Theory of the N-Body Problem
June 9, 1996
19
The simplest method of extrapolating integration is called Euler's method, and
when applied to star movement with time broken down into steps of size h, the result
comes out as:
That is, at time
t+h
, the velocity will be equal to the velocity at time
t
, plus how much the
star has accelerated over the time period, assuming that the acceleration is constant. Simi-
larly, the new position is based on the original position and the velocity.
Equation (EQ 2) can be derived from (EQ 1) as follows:
As FIGURE 14. shows, Euler's method can result in large errors because the accel-
eration
isn't
constant, it changes over the time period. Taylor's theorem says that the exact
solution will really have the form:
v t h
+
(
)
v t
( )
h a t
( )
+
=
x t h
+
(
)
x t
( )
h v t
( )
+
=
(EQ 2)
x t
( )
f x t
( )
(
)
t t
d
d
=
v t
( )
f x t
( )
(
)
t
d
=
v t h
+
(
)
f x
( )
(
)
d
-
t
f x
( )
(
)
d
tt h
+
+
=
v t h
+
(
)
v t
( )
h f
x t
( )
(
)
+
v t h
+
(
)
v t
( )
f x
( )
(
)
d
tt h
+
+
=
f(x(t))
v(t)+hf(x(t))
v(t)
t
t+h
exact velocity
FIGURE 14. Graph of Euler's method
t
v
v t h
+
(
)
v t
( )
ha t
( )
12
h
2
a
'
( )
+
+
=