Mathematical possibilities of Newton's Third Law

 $$\textbf{F}(\textbf{r}_1,\textbf{r}_2)=\textbf{F}(\textbf{r}_2,\textbf{r}_1)$$

The simplest possibilities are 

$$r=|\textbf{r}_1-\textbf{r}_2|$$$$\textbf{r}=\textbf{r}_1+\textbf{r}_2$$

Sub-lattice pairing in Graphene

$$E=mv_F^2$$

Low-speed Relativistic Phenomena

 If we have macroscopic quantum phenomena, why not low-speed relativistic phenomena?!

In fact, the first example already exists: Graphene! which has

$$E=pv_F,$$

where \(v_F\) is the Fermi velocity, and \(p=\hbar k\). This suggests that \(v_F\) plays the role of \(c\)

$$\gamma_F=\frac{1}{\sqrt{1-(\frac{v}{v_F})^2}}.$$