Two body system in Newtonian gravity

[emptymalei]

Depends on

• Central force problem #13

Definition of a Two Body System

Two masses $m_1$ and $m_2$.

We use Jacobi coordinates, $\vec R=\frac{m_1 \vec x_1 + m_2 \vec x_2}{m_1 + m_2}$, $\vec r = \vec x_1 - \vec x_2$.

The equation we are solving is

$$ \ddot {\vec r} = - \frac{\alpha}{r^2} \hat r, $$

where $\alpha=G(m_1+m_2)$.

Simplify

We reduce the problem to $R=0$,

$$ m_1 x_1 + m_2 x_2 = 0. $$

The displacement $r$ becomes

$$ \hat r = \left( 1 + \frac{m_1}{m_2} \right) \hat x_1. $$

Design

1. Limit the params.
2. Which coordinate to use.

References:

1. https://en.wikipedia.org/wiki/Two-body_problem

[cmp0xff]

I would go with

1. Radial motion of one body in one dimension
2. Circular motion of one body in two dimensions, inheriting the previous result
3. Optionally circular motion of one body in three dimensions, inheriting the previous result

Then go up to to bodies.

[emptymalei]

@cmp0xff agree. This issue depends on #13