Central Force and Reduction of Two body problem into One body problem

Central Force:

The force which is always directed towards or away from a fixed point. For eg: gravitational force, magnetic force, etc.
This force is the function of the distance of a point from the fixed point.
The central force acting along \(\vec{r}\) is expressed as
\begin{align*}\vec{F}=\hat{r}f(r) \end{align*}\begin{align*}where,\space \hat{r}=unit\space vector\space along\space \vec{r} \end{align*}\begin{align*}f(r)=magnitude\space of\space the\space force \end{align*}

Two body problem reduced to one body problem:

A two body problem involving central forces can always be reduced to the form of a one body problem thereby simplifying the calculation.
Consider two particles of masses \(m_1\) and \(m_2\) with instantaneous position vectors \(\vec{r_1}\) and \(\vec{r_2}\) respectively with respect to origin O. The vector distance of particle \(m_2\) from \(m_1\) is
\(\vec{r}=\vec{r_2}-\vec{r_1}\)
The central force acting on\(m_1\) due to presence of\(m_2\) is
\(\vec{F_{12}}=m_1\frac{d^{2}\vec{r_1}}{dt^{2}}\)->eqn I
Similarly, force acting on\(m_2\) due to\(m_1\) is
\(\vec{F_{21}}=m_2\frac{d^{2}\vec{r_2}}{dt^{2}}\)->eqn II
These forces are acting along the line joining\(m_1\) and \(m_2\).
Since, there is no external force acting on the system,
\begin{align*}F_{12}=-\hat{r}f(r) \end{align*}\begin{align*}m_1\frac{d^{2}\vec{r_1}}{dt^{2}}=-\frac{\hat{r}f(r)}{m_1}->eqn\space III \end{align*}Again,\begin{align*}F_{21}=\hat{r}f(r) \end{align*}\begin{align*}m_2\frac{d^{2}\vec{r_2}}{dt^{2}}=\frac{\hat{r}f(r)}{m_2}->eqn\space IV \end{align*}Subtracting eqn(III) from eqn(IV), we get\begin{align*}\frac{d^{2}\vec{r_2}}{dt^{2}}-\frac{d^{2}\vec{r_1}}{dt^{2}}=\hat{r}{\frac{1}{m_1}+\frac{1}{m_2}}f(r) \end{align*}\begin{align*}d \frac{d^{2}(\vec{r_2}-\vec{r_1})}{dt^{2}}=\frac{\hat{r}f(r)}{\mu }\end{align*}\begin{align*}\mu \frac{d^{2}\vec{r}}{dt^2}=\hat{r}f(r) \end{align*}where \begin{align*}\frac{1}{\mu}=\frac{1}{m_1}+\frac{1}{m_2}\end{align*}\begin{align*}\mu=\frac{m_1.m_2}{m_1+m_2}is\space reduced\space of\space the\space system\end{align*}\begin{align*}or,\space \vec{F}=\mu \frac{d^{2}\vec{r}}{dt^2} \end{align*}\begin{align*}\vec{F}=\mu \vec{a} \end{align*}Equation of motion of a particle having mass \(\mu \). Hence, two body problem is reduced to a single body problem.
Reduced mass of a system of particles is the fictitious mass of the system observed by an observer. The introduction of \(\mu \) in physical relations can help reduce a many body system into a simpler body system. The reduced mass of a system is always less than the smallest mass of the system.

References

Adhikari, Pitri Bhakta. A Textbook of Physics Volume-I. Kathmandu: Sukunda Pustak Bhawan, 2015.

Feynman, Richard P. The Feynman Lectures on Physics Volume 1. Noida: Dorling Kindersley (India) Pvt. Ltd., 2014.

Mathur, D S. Mechanics. New Delhi: S. Chand & Company Pvt. Ltd., 2015.

Young, Hugh D, Roger A Freedman and A Lewis Ford. University Physics. Noida: Dorling Kindersley (India) Pvt. Ltd., 2014.