Overview of the Two-body Problem

Here I will present the exact solution of the two body problem.

Definition of termsPermalink

First we define the center of mass coordinate, $R$:

As $m_1$ and $m_2$ are invariant, we can express $P$, the total momentum, in terms of $\dot{R}$:

How does the center of mass move?Permalink

As total momentum is conserved, we have (in scalars) $\dot{R}=\frac{P}{m_1+m_2}$.

Hence, we have

Therefore Center-of-Mass is always on a straight line with constant velocity!

Solving the SystemPermalink

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:

Further differentiating yields:

Hence we have:

Leading to:

Where

is named the reduced mass. Note that it can be written as:

Concluding RemarksPermalink

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 ReaderPermalink

• 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