For a system of particles we have the mutual forces among the selected particles referred to as internal forces, otherwise external forces. If there are no external forces, the system is called closed, otherwise open.
> *Definition 1*: the internal interaction forces $\mathbf{F}_i$ in a system of $n \in \mathbb{N}$ particles with position $\mathbf{x}_i$ may be approximated by pairwise interaction forces given by
> for all $\mathbf{x}_i$ with $\mathbf{F}_{ij}$ the pairwise interaction force between particle $i$ and $j$.
For high density systems this approximation diverges.
## Systems with conservative internal forces
Considering a system of $n \in \mathbb{N}$ particles with position $\mathbf{x}_i$ and mass $m_i \in \mathbb{R}$ with conservative external forces $\mathbf{F}_i$. For each particle an equation of motion can be formulated using the pairwise interaction approximation (definition 1), obtaining
> with $\mathbf{F}_i$ the conservative external force.
<br>
> *Proposition 1*: the total momentum $\mathbf{P}: t \mapsto \mathbf{P}(t)$ is related to the total external force $\mathbf{F}: t \mapsto \mathbf{F}(t)$ by
>
> $$
> \mathbf{P}'(t) = \mathbf{F}(t),
> $$
>
> for all $t \in \mathbb{R}$.
??? note "*Proof*:"
Will be adder later.
> *Proposition 2*: the total angular momentum $\mathbf{J}: t \mapsto \mathbf{J}(t)$ is related to the total external torque $\mathbf{\Gamma}: t \mapsto \mathbf{\Gamma}(t)$ by
>
> $$
> \mathbf{J}'(t) = \mathbf{\Gamma}(t),
> $$
>
> for all $t \in \mathbb{R}$ if the internal forces are central forces.
??? note "*Proof*:"
Will be adder later.
### Orbital and spin angular momentum
Considering internal position vectors $\mathbf{r}_i$ relative to the center of mass $\mathbf{r}_i = \mathbf{x}_i - \mathbf{R}$. I propose that the total angular momentum $\mathbf{J}$ can be expressed as a superposition of the orbital $\mathbf{L}$ and spin $\mathbf{S}$ angular momentum components given by
$$
\mathbf{J} = \mathbf{L} + \mathbf{S}.
$$
??? note "*Proof*:"
Will be added later.
> *Definition 6*: the orbital angular momentum $\mathbf{L}$ of the system is defined as
>
> $$
> \mathbf{L} = \mathbf{R} \times \mathbf{P},
> $$
>
> with $\mathbf{R}$ the center of mass and $\mathbf{P}$ the total momentum of the system.
<br>
> *Definition 7*: the spin angular momentum $\mathbf{S}: t \mapsto \mathbf{S}(t)$ of the system is defined as
> *Proposition 3*: let $\mathbf{L}: t \mapsto \mathbf{L}(t)$ be the orbital angular momentum and let $\mathbf{S}: t \mapsto \mathbf{S}(t)$ be the spin angular momentum. Then we have
>
> $$
> \begin{align*}
> \mathbf{L}'(t) &= \mathbf{\Gamma}_o(t), \\
> \mathbf{S}'(t) &= \mathbf{\Gamma}_s(t),
> \end{align*}
> $$
>
> for all $t \in \mathbb{R}$ with $\mathbf{\Gamma}_o: t \mapsto \mathbf{\Gamma}_o(t)$ and $\mathbf{\Gamma}_s: t \mapsto \mathbf{\Gamma}_s(t)$ the orbital and spin torque.
### Energy
> *Definition 9*: the total kinetic energy $T$ of the system is defined as
>
> $$
> T = \sum_{i=1}^n \frac{1}{2} m_i \|\mathbf{x}_i'\|^2,
> $$
>
> with $\mathbf{x}_i$ the position of each particle.
<br>
> *Definition 10*: the orbital and internal kinetic energy $T_{o,r}$ of the system are defined as