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Twobody problem
An isolated dynamical system consisting of two freely moving point objects exerting forces on one another is conventionally termed a twobody problem.
Suppose that the first object is of mass
and is located
at position vector
. Likewise, the second object
is of mass
and is located at position vector
.
Let the first object exert a force
on the
second. By Newton's third law, the second object exerts an
equal and opposite force,
, on the
first. Suppose that there are no other forces in the problem. The equations of motion of our two objects are thus
where
.
Figure 2.7:
Twobody problem.

Now, the center of mass of our system is located at

(2.73) 
Hence, we can write
where
. (See Figure 2.7.)
Substituting the preceding two equations into Equations (2.71) and (2.72),
and making use of the fact that the center of mass of an isolated system
does not accelerate (see Section 2.5), we find that both equations yield

(2.76) 
where

(2.77) 
is called the reduced mass. Hence, we have effectively converted our
original twobody problem into an equivalent onebody problem. In the equivalent problem, the
force
is the same as that acting on both objects in the original problem (except for a minus sign); however, the mass,
, is different, and
is less than either of
or
(which is why it is called the ``reduced'' mass).
We conclude that the dynamics of an isolated system consisting of two interacting point objects can
always be reduced to that of an equivalent system consisting of a single point object moving in a fixed force field.
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Richard Fitzpatrick
20160331