Question

How can we change two body system into one.

Answer 1

Suppose we have a two body system (body 1, body 2) with a potential depending only on their relative distance. The total energy is :

$E = \frac{m_1 v_1^2}{2} + \frac{m_2 v_2^2}{2} + V(r_1-r_2)$

and thus we introduce the relative position $r=r_1-r_2$, position of a center of mass $R=\frac{m_1 r_1 + m_2 r_2}{m_1+m_2}$ and we reformulate the positions $r_1, r_2$ with these new variables :

$r_1 = R+\frac{m_2}{m_1+m_2} r$

$r_2 = R - \frac{m_1}{m_1+m_2}r$

And now the velocities :

$v_1 = v_{MC} + \frac{m_2}{m_1+m_2} v$

$v_2 = v_{MC} - \frac{m_1}{m_1+m_2} v$

Now by reformulating the energy in these new variables we get :

$E =\frac{(m_1+m_2)v_{MC}^2}{2} + \frac{1}{2} \frac{m_1m_2}{m_1+m_2} v^2+V(r)$

We see that the potential energy is independent from R and thus $v_{MC}$ is conserved (which is coherent with the physical explanation - the system is isolated and thus the total momentum is conserved). We can place ourselves in a inertial frame of reference where $v_{MC}=0$. Now by introducing $\mu=\frac{m_1 m_2}{m_1 + m_2}$ - reduced mass, we get an energy of a one body in a central field :

$E = \frac{\mu v^2}{2} +V(r)$

And now we can solve the problem completely in this formalism.