Problems with two interacting objects can be simplified
by combining them into a pseudo-object with reduced mass ,
whose position equals the relative position of the objects.
For bodies 1 and 2 with respective masses and :
If and are the objects’ respective positions,
then we define
the relative position ,
the relative velocity ,
and the relative acceleration :
We now choose the coordinate system’s origin
to be the center of mass of both objects:
Rearranging and differentiating then yields the following useful equations:
Using these relations, we can rewrite the relative quantities we defined earlier:
Meanwhile, Newton’s third law states that
if object 1 experiences a force caused by object 2,
then object 2 experiences an opposite and equal force .
In fact, our earlier relation between and
boils down to Newton’s third law:
With all that in mind, let us take a closer look at the relative acceleration :
Where is the reduced mass, as defined above.
In other words, the relative acceleration
is just multiplied by .
This can be regarded as focusing on the dynamics of body 1,
while correcting for the effects of body 2.
This also suggests the following way
to recover the original positions and
from , which you can easily verify for yourself:
With this, we can rewrite the total kinetic energy in an elegant way:
Then, assuming that the system’s potential energy
only depends on the distance between the two objects,
we just showed that we can rewrite both and
to contain only and relative quantities.
This is relevant for both Lagrangian mechanics
and Hamiltonian mechanics,
where and respectively.