In this post, we primarily elucidate the equations of motion for multiple point masses and the constraints associated therewith。
The equation of motion (Newton’s second law) for the \(i\)th particle is written as \[ \sum_j \mathbf{F}_{j i}+\mathbf{F}_i^{(e)}=\dot{\mathbf{p}}_i, \tag{1.1} \] where \(\mathbf{F}_i^{(e)}\): external force, \(\mathbf{F}_{j i}\): internal force on the \(i\)th particle due to the \(j\)th particle ( \(\mathbf{F}_{i i}=0\)), \(\mathbf{p}\): linear momentum.
Summed over all particles \[ \frac{d^2}{d t^2} \sum_i m_i \mathbf{r}_i=\sum_i \mathbf{F}_i^{(e)}+\sum_{\substack{i, j \\ i \neq j}} \mathbf{F}_{j i} .\tag{1.2} \] the average radii vector \(\mathbf{R}\) \[ \mathbf{R}=\frac{\sum m_i \mathbf{r}_i}{\sum m_i}=\frac{\sum m_i \mathbf{r}_i}{M}.\tag{1.3} \] The \(\mathbf{R}\) defines a point also known as the center of mass, with this definition, (1.2) reduces to \[ M \frac{d^2 \mathbf{R}}{d t^2}=\sum_i \mathbf{F}_i^{(e)} \equiv \mathbf{F}^{(e)},\tag{1.4} \] total momentum of the system \[ \mathbf{P}=\sum m_i \frac{d \mathbf{r}_i}{d t}=M \frac{d \mathbf{R}}{d t},\tag{1.5} \] total angular momentum of system \[ \sum_i\left(\mathbf{r}_i \times \dot{\mathbf{p}}_i\right)=\sum_i \frac{d}{d t}\left(\mathbf{r}_i \times \mathbf{p}_i\right)=\dot{\mathbf{L}}=\sum_i \mathbf{r}_i \times \mathbf{F}_i^{(e)}+\sum_{\substack{i, j \\ i \neq j}} \mathbf{r}_i \times \mathbf{F}_{j i}\tag{1.6} \]
\(\mathbf{F}_{j i}=\mathbf{F}_{i j}\)