For a system of two interacting particles 1, 2 we get from the conservation of momentum
$$ \dot{\bf{p_1}} + \dot{\bf{p_2}} = 0$$
and from conservation of angular momentum
$$ \bf{r_1} \times \dot{\bf{p_1}} + \bf{r_2} \times \dot{\bf{p_2}} = 0$$
so
$$ \bf{F_1} = -\bf{F_2} $$
and
$$ \bf{r_{12}} \times \bf{F_1} = 0 $$
where $ \bf{r_{12}} = \bf{r_1} - \bf{r_2} $ is the separation between particles. So basically only central forces $\bf{F}=\bf{F}\left(|\bf{r_{12}}|\right)$ are allowed.
But for a system of three and more interacting particles we don't have such restrictions. E.g. for 3 particles
$$ \bf{F_3} = -\bf{F_1}-\bf{F_2}$$
and
$$ \bf{r_{13}} \times \bf{F_1} + \bf{r_{23}} \times \bf{F_2} = 0 $$
(two more likewise expressions for (21,31) and (12,32)). And this is, I believe, as far as one can get in general case.
We can expand $$ \bf{F_1} = \bf{F_{1,2}} + \bf{F_{1,3}} + \bf{R_1} $$ where $\bf{F_{1,2}}$ and $\bf{F_{1,3}}$ are for two-body interactions, but no one forbids the existence of 3-body force $\bf{R_1}\left(\bf{r_{12}}, \bf{r_{23}}, \bf{r_{31}}\right)$.
Are there any known examples or I am missing something?
