mechanics of a system of particles

In this post, we primarily elucidate the equations of motion for multiple point masses and the constraints associated therewith。

question

4-5 min

regular

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 \[ \sum_i\left(\mathbf{r}_i \times \dot{\mathbf{p}}_i\right)=\sum_i \frac{d}{d t}\left(\mathbf{r}_i \times \mathbf{p}_i\right)=\dot{\mathbf{L}}=\sum_i \mathbf{r}_i \times \mathbf{F}_i^{(e)}+\sum_{\substack{i, j \\ i \neq j}} \mathbf{r}_i \times \mathbf{F}_{j i}\tag{1.6} \]

\(\mathbf{F}_{j i}=\mathbf{F}_{i j}\)