Overview of the Two-body Problem
Here I will present the exact solution of the two body problem.
Definition of terms
First we define the center of mass coordinate, \(R\):
\[\underline{R}=\frac{m_1 \underline{r_1}+m_2 \underline{r_2}}{m_1+m_2}\]As \(m_1\) and \(m_2\) are invariant, we can express \(P\), the total momentum, in terms of \(\dot{R}\):
\[\underline{P}=(m_1+m_2)\dot{R}\]How does the center of mass move?
As total momentum is conserved, we have (in scalars) \(\dot{R}=\frac{P}{m_1+m_2}\).
Hence, we have
\[\underline{R}(t)=\underline{R}(0)+\frac{\underline{P}}{m_1+m_2} t.\]Therefore Center-of-Mass is always on a straight line with constant velocity!
Solving the System
Now we assume the two bodies interact via some conservative force that depends on the relative coordinate \(\mathbf{r} = \mathbf{r_1} - \mathbf{r_2}\), namely \(V = V(\mathbf{r_1} - \mathbf{r_2})\).
By putting into the expression of \(\underline{R}\), we obtain:
\[\underline{r}_1=\underline{R}+\frac{m_2}{m_1+m_2} \underline{r}\]Further differentiating yields:
\[\underline{\ddot{r}}_1=\frac{m_2}{m_1+m_2}{\underline{\ddot{r}}}\]Hence we have:
\[\frac{m_1 m_2}{m_1+m_2} {\underline{\ddot{r}}}=m_1 {\underline{\ddot{r}}}_1=-\underline{\nabla}_1 V\left(\underline{r}_1-\underline{r}_2\right)=-\underline{\nabla} V(\underline{r})\]Leading to:
\[m {\underline{\ddot{r}}}=-\underline{\nabla} V(\underline{r})\]Where
\[m \equiv \frac{m_1 m_2}{m_1+m_2}\]is named the reduced mass. Note that it can be written as:
\[\frac{1}{m}=\frac{1}{m_1}+\frac{1}{m_2}.\]Concluding Remarks
N-body problem is one of the most important problems in physics! (as bodies can be particles, planets, anything!) I will do a supervised project (Summer 2024) with Dr Jenni Smillie, focusing on three-body gravitational problems. To know more about my research progress, please press the “Research” tag below!
Exercises for the Reader
- From \(\dot{R}=\frac{P}{m_1+m_2}\), show that \(\underline{R}(t)=\underline{R}(0)+\frac{\underline{P}}{m_1+m_2} t\). (Difficulty: F4-5 HKDSE)
- Give the definition for \(\nabla\). Explain the difference between \(\nabla\) and \(\nabla_1\). (Difficulty: University Year 1-2)
- What is the form of the reduced mass similar to? (Hint: in circuits) (Difficulty: F5 HKDSE)
- Show that two-body motion is planar (Difficulty: University Year 1-2)
Last Updated - 3/6/2024
