Rigid bodies and systems of particles¶

Consider a set of particles which each has a mass \(m_i\), and position vector \(\vec{r}_i\), then each particle can experience two forces:

\(\vec{f}~{int}\) — internal forces which act between particles in a system,
\(\vec{f}~{ext}\) — external forces which act on the system from outside, e.g. a magnetic flux.

The overall motion of a system is disjoint from its internal motions, thanks to Newton’s third law, since \(F_{ij} = - F_{ji}\), where \(F_{ab}\) is the force exerted on particle \(a\) by particle \(b\), and the force \(F_{ij} \propto (\vec{r}_i - \vec{r}_j)\), i.e. acts upon the shortest path between the particles.

The Centre of Mass¶

The total mass of a system of particles is

\[ \begin{align}\begin{aligned} \label{eq:5} M = \sum_{i=1}^N m_i\\from which the total momentum can be found as\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned} \label{eq:6} \vec{P} = \sum_{i=1}^N \vec{p}_i = \sum_{i=1}^N \dot{\vec{r}}_i m_i\\which, defining the centre of mass,\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}\vec{R} = \frac{\sum_i m_i \vec{r}_i}{\sum_i m_i}\\leads to\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned} \label{eq:7} \vec{P} = M \dv{t} \vec{R}\\The force on the :math:`i`\ th particle is then\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned} \label{eq:8} F_i = F_i^{\rm (ext)} + \sum_{j \neq i} \vec{F}_{ji} = \dv{t}\vec{P}\\so\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}\dv{t}\vec{P} = \sum_i \qty[ \vec{F}_i^{(\rm ext)} + \sum_{j \neq i} \vec{F}_{ji}]\\We have\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}\begin{split} \begin{aligned} \sum_i \sum_{j \neq i} \vec{F}_{ji} &= \sum_i \sum_{j < i} \vec{F}_{ji} + \sum_i \sum_{j > i} \vec{F}_{ji} \\ &= \sum_i \sum_{j<i} (\vec{F}_{ji} + F_{ij}) = 0\end{aligned}\end{split}\\Let :math:`\vec{F}^{(\rm ext)} = \sum_i \vec{F}_i^{\rm (ext)}`, then\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned} \label{eq:9} F^{\rm (ext)} = \dv{t} P = M \dv[2]{t} \vec{R}\\Thus, the overall motion of a system relies only on the external force\end{aligned}\end{align} \]

acting upon it.

Angular Momentum¶

We have

\[ \begin{align}\begin{aligned}m_i \dv[2]{t} r_i = \vec{F}_i^{\rm (ext)} + \sum_{j \neq i} \vec{F}_{ji}\\and the total angular momentum of a system as\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned} \label{eq:10} \vec{L} = \sum^N_{i=1} \vec{r}_i \cp \vec{p}_i = \sum_{i=1}^N m_i \vec{r}_i \cp \dot{\vec{r}}_i\\Then\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}\begin{split} \begin{aligned} \dot{\vec{L}} & = \dv{t} \sum_{i=1}^N m_i \vec{r}_i \cp \dot{\vec{r}} \nonumber \\ & = \sum_{i=1}^N m_i \dv{t} (\vec{r}_i \cp \dot{\vec{r}} )\nonumber \\ & = \sum_{i=1}^N m_i \vec{r}_i \cp \ddot{\vec{r}}_i \nonumber \\ & = \sum_{i=1}^N \vec{r}_i \cp \vec{F}_i^{\rm (ext)} + \sum_{i=1}^N \sum_{j \neq i} \vec{r}_i \cp \vec{F}_{ji} \nonumber \\ & = \vec{G}^{\rm (ext)} + \sum_{i=1}^N \sum_{j>i} (\vec{r} \cp \vec{F}_{ji} + \vec{r}_j \cp \vec{F}_{ij} ) \nonumber \\ & = \vec{G}^{\rm (ext)} + \sum_{i=1}^N \sum_{j>i} (\vec{r}_i - \vec{r}_j) \cp \vec{F}_{ji} \nonumber \\ \therefore \dot{\vec{L}} & = \vec{G}^{\rm (ext)}\end{aligned}\end{split}\\Thus angular momentum is conserved unless there is an applied torque.\end{aligned}\end{align} \]

Separation of Kinetic Energy¶

Let the position of a particle be described relative to the centre of mass, i.e.

\[ \begin{align}\begin{aligned}\vec{r}_i = \vec{R} + \vec{r}'_i\\Then\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}\begin{split} \begin{aligned} \sum_{i=1}^N m_i \vec{r}'_i &= \sum^N_{i=1} m_i \vec{r}_i - \sum_{i=1}^N m_i R \\ &= M \qty[ \frac{\sum m_i \vec{r}_i}{\sum m_i } - \vec{R}] = 0\end{aligned}\end{split}\\the kinetic energy :math:`T` is then\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}\begin{split} \begin{aligned} T &= \half \sum m_i \dv{\vec{r}_i}{t}^2 \nonumber\\ &= \half \sum_{i=1}^N m_i \qty[ \dot{R}^2 + 2 \dot{\vec{r}}'_i \vdot \dot{\vec{R}} + (\dot{\vec{r}}'_i)^2 ] \nonumber\\ &= \half \sum m_i \dot{\vec{R}}^2 + \half \sum m_i \dot{\vec{r}'_i}^2 + \sum m_i \dot{\vec{r}_i'} \nonumber\\ &= \half \sum m_i \dot{\vec{R}}^2 + \half \sum m_i (\dot{\vec{r}}'_i)^2\end{aligned}\end{split}\\Thus the kinetic energy is the sum of the internal energies and the\end{aligned}\end{align} \]

kinetic energy of a single particle with the mass of the whole system.

Separation of Angular Momentum¶

The total angular momentum of a system is

\[ \begin{align}\begin{aligned}\begin{split} \begin{aligned} \vec{L} &= \sum \vec{r}_i \cp \vec{p}_i \nonumber \\ &= \sum m \vec{r}_i \cp \dot{\vec{r}}_i \nonumber\\ &= \sum m_i (\vec{R} + \vec{r}_i') \cp (\dot{\vec{R}} + \dot{\vec{r'}}_i ) \nonumber\\ &= M \vec{R} \cp \dot{\vec{R}} + \qty[ \sum m_i \vec{r}' ] \cp \dot{\vec{R}} \nonumber\\ & \quad+ \vec{R} \cp \qty[ \sum m_i \dot{\vec{r}}'_i ] + \sum m_i \vec{r}'_i \cp \dot{\vec{r}}'_i \nonumber\\ &= M \vec{R} \cp \dot{\vec{R}} + \vec{L}~{int}\end{aligned}\end{split}\\Where :math:`\vec{L}~{int} = \sum m_i \vec{r}'_i \cp \dot{\vec{r}}'_i`,\end{aligned}\end{align} \]

\[\begin{split}\begin{aligned} \dot{\vec{L}} &= \sum r_i \cp \vec{F}_i^{\rm (ext)} \nonumber\\ &= \underbracket{\vec{R} \cp \sum \vec{F}_i^{\rm (ext)}}_{\text{torque on system}} + \sum \underbracket{\vec{r}'_i \cp \vec{F}_i^{\rm (ext)}}_{\text{torque on each particle}}\end{aligned}\end{split}\]