reduction from two-body to one-body problem

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Reduced mass and center of mass coordinates

If the potential energy of two bodies depends only on their separation r =|\mathbf{r}_1 - \mathbf{r}_2|\,, then it is convenient to introduce new generalized coordinates, namely the separation vector

\mathbf{r} = \mathbf{r}_1 - \mathbf{r}_2\,,

and the center of mass

\mathbf{R}_{com} = \frac{m_1 \mathbf{r}_1 + m_2 \mathbf{r}_2}{m_1 + m_2}\,.

Equivalently,

\mathbf{r}_1 = \mathbf{R}_{com} + \frac{m_2}{m_1+m_2} \mathbf{r}\,,
\mathbf{r}_2 = \mathbf{R}_{com} - \frac{m_1}{m_1+m_2} \mathbf{r}\,.

Note that

\frac{1}{2} m_1 v_1^2 = \frac{1}{2} m_1 \left( V_{com}^2 + 2 \frac{m_2}{m_1+m_2} \mathbf{R}_{com}\cdot\mathbf{r} + \frac{m_2^2}{(m_1+m_2)^2} v^2 \right)\,,

so that

\frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_1 v_1^2 = \frac{1}{2}(m_1+m_2) V_{com}^2 + \frac{1}{2} \frac{m_1 m_2}{m_1 + m_2} v^2\,,

and

T = \frac{1}{2} M V_{com}^2 + \frac{1}{2} \mu v^2\,,

where M = m_1+m_2\, is the total mass while

\mu = \frac{m_1 m_2}{m_1+m_2}\,,

is the reduced mass.

Lagrangian

In terms of these new generalized coordinates, the Lagrangian can be written

L = \frac{1}{2} M \dot\mathbf{R}_{com}^2 + \frac{1}{2} \mu \dot\mathbf{r}^2 - V(r)\,.

Since \mathbf{R}\, is cyclic, we may choose our reference frame such that \dot\mathbf{R}_{com} = \mathbf{R}_{com} = 0\,. Then

L = \frac{1}{2} \mu \dot\mathbf{r}^2 - V(r)\,.

Since central force motion is planar, we may parameterize the plane using r\, and \phi\,, so that

\dot\mathbf{r} = \dot r \hat\mathbf{r} + r \dot\phi \hat\boldsymbol{\phi}\,.

(See kinematics in spherical coordinates). Then

L = \frac{1}{2} \mu \dot r^2 + \frac{1}{2}\mu r^2 \dot\phi^2 - V(r)\,.

Angular momentum

In the center of mass frame, the angular momentum is simply

\mathbf{l} = \mathbf{r}\times (\mu \dot\mathbf{r}) = m r^2 \dot\phi \hat\mathbf{z}\,

where \hat\mathbf{z}\, is perpendicular to the plane, so that \mathbf{l}\, has magnitude

l = m r^2 \dot\phi\,.

We use the symbol l\, instead of L\, to distinguish the angular momentum from the Lagrangian.

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