 Zero-natural-length spring

An ideal spring with stiffness $k>0$ and zero natural length has extension equal to the distance between its endpoints. Its elastic potential energy is exactly $k|{\mathbf{r}}_2-{\mathbf{r}}_1{|}^2/2$, and its force on endpoint 2 is $-k({\mathbf{r}}_2-{\mathbf{r}}_1)$. Thus its Cartesian stiffness is isotropic and constant. Coupled systems of such springs have exactly linear displacement equations after a constant force, such as gravity, is absorbed into the equilibrium position. This special model differs from an ordinary finite-natural-length spring, whose transverse small-motion stiffness depends on its tension and equilibrium length.