In short, I'm trying to prove that it is possible for a mechanical system to stay in equilibrium under some external forces (with the "right" internal forces) if and only if the net force and the net torque they produce are both $0$. This translates to the following linear algebra problem.
Let $\mathbf{x}_{ij}\in \mathbb{R}^3$ be variable vectors for $1\leq i,j\leq n$, $i\neq j$ (representing internal forces). Also let $\mathbf{f}_i, \mathbf{r}_i\in\mathbb R^3$ be given vectors for $1\leq i\leq n$ (representing net forces and radius vectors for each body). Consider the following system of linear equations:
- $\sum _{j:j\neq i} \mathbf{x}_{ij}=\mathbf{f}_i$ for every $i$.
- $\mathbf{x}_{ij}+\mathbf{x}_{ji}=0$ for every $i\neq j$.
- $(\mathbf{r}_i - \mathbf{r}_j)\times \mathbf{x}_{ij}=0$ for every $i\neq j$ (this is the cross product).
I want to show that this system has a solution if and only if $\sum_i \mathbf{f}_i = 0$ and $\sum_i \mathbf{r}_i\times \mathbf{f}_i=0$.
The $\Rightarrow$ direction is easy. For the other direction, I worked out some simple cases by hand but I can't see an elegant way to reason about the general case.