- #1

Arman777

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I am trying to create a simulation for a gravitational 2 body problem.

But I am kind of having trouble to define the equations that can be solve numerically. From an inertial frame I defined the position of the two objects as the ##\vec{r_1}## and ##\vec{r_2}## with masses ##m_1## and ##m_2##.

Let the ##\vec{R}_{CM}## be the position of the CM of the objects. Now from the perspective of the CM, we can write position vectors of the objects in terms of ##\vec{r'}_1## and ##\vec{r'}_2##.

$$\vec{r'}_1 = \frac{-m_2}{m_1 + m_2} \vec{r}~~(1)$$

and $$\vec{r'}_2 = \frac{m_1}{m_1 + m_2} \vec{r}~~(2)$$where

##\vec{r}= \vec{r'}_2 - \vec{r'}_1##

Now in this case we can use the reduced mass and define the force on this mass. So we have,

##\vec{F} = \mu \ddot{\hat{r}} = -\frac{Gm_1m_2}{r^2} \hat{r}##

Now I need to solve this equation and put back into the (1) and (2) right ?

But I am kind of having trouble to define the equations that can be solve numerically. From an inertial frame I defined the position of the two objects as the ##\vec{r_1}## and ##\vec{r_2}## with masses ##m_1## and ##m_2##.

Let the ##\vec{R}_{CM}## be the position of the CM of the objects. Now from the perspective of the CM, we can write position vectors of the objects in terms of ##\vec{r'}_1## and ##\vec{r'}_2##.

$$\vec{r'}_1 = \frac{-m_2}{m_1 + m_2} \vec{r}~~(1)$$

and $$\vec{r'}_2 = \frac{m_1}{m_1 + m_2} \vec{r}~~(2)$$where

##\vec{r}= \vec{r'}_2 - \vec{r'}_1##

Now in this case we can use the reduced mass and define the force on this mass. So we have,

##\vec{F} = \mu \ddot{\hat{r}} = -\frac{Gm_1m_2}{r^2} \hat{r}##

Now I need to solve this equation and put back into the (1) and (2) right ?

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