Question

$ddot{mathbf{r}}_1 = -gm_2 \frac{mathbf{r}_1 - mathbf{r}_2}{|mathbf{r}_1 - mathbf{r}_2|^3} - gm_3 \frac{mathbf{r}_1 - mathbf{r}_3}{|mathbf{r}_1 - mathbf{r}_3|^3}$
$ddot{mathbf{r}}_2 = -gm_1 \frac{mathbf{r}_2 - mathbf{r}_1}{|mathbf{r}_2 - mathbf{r}_1|^3} - gm_3 \frac{mathbf{r}_2 - mathbf{r}_3}{|mathbf{r}_2 - mathbf{r}_3|^3}$
$ddot{mathbf{r}}_3 = -gm_1 \frac{mathbf{r}_3 - mathbf{r}_1}{|mathbf{r}_3 - mathbf{r}_1|^3} - gm_2 \frac{mathbf{r}_3 - mathbf{r}_2}{|mathbf{r}_3 - mathbf{r}_2|^3}$
$mathbf{r}_1 - mathbf{r}_2$, $mathbf{r}_1 - mathbf{r}_3$, $mathbf{r}_2 - mathbf{r}_3$, $mathbf{r}_1$, $mathbf{r}_2$, $mathbf{r}_3$, $mathbf{r}_2 - mathbf{r}_3$

Explanation:

Step1: Identify the Discipline

The equations involve gravitational forces and motion of celestial bodies (or masses), which falls under Physics (a subfield of Natural Science). The variables like \( \ddot{\mathbf{r}} \) (acceleration vectors), \( G \) (gravitational constant), \( m \) (masses), and position vectors \( \mathbf{r} \) indicate a physics problem related to celestial mechanics or the n - body problem (specifically the three - body problem here).

Step2: Analyze the Equations

The equations are the equations of motion for a three - body system under gravitational interaction. The general form of the gravitational force between two masses \( m_i \) and \( m_j \) is \( \mathbf{F}_{ij}=-Gm_im_j\frac{\mathbf{r}_i - \mathbf{r}_j}{|\mathbf{r}_i-\mathbf{r}_j|^3} \), and by Newton's second law \( \mathbf{F}=m\ddot{\mathbf{r}} \), we get the acceleration of each mass as the sum of the gravitational forces from the other two masses. For example, for mass \( m_1 \), the acceleration \( \ddot{\mathbf{r}}_1 \) is the sum of the gravitational accelerations due to \( m_2 \) and \( m_3 \), which is \( \ddot{\mathbf{r}}_1=-Gm_2\frac{\mathbf{r}_1 - \mathbf{r}_2}{|\mathbf{r}_1-\mathbf{r}_2|^3}-Gm_3\frac{\mathbf{r}_1 - \mathbf{r}_3}{|\mathbf{r}_1-\mathbf{r}_3|^3} \), and similarly for \( \ddot{\mathbf{r}}_2 \) and \( \ddot{\mathbf{r}}_3 \). The diagram also shows the position vectors and the relative position vectors between the three masses, which is consistent with the three - body problem in physics.

Answer:

The subfield is Physics (under Natural Science), related to the three - body problem and gravitational dynamics.