= Zero-natural-length spring
{title2=$V=\tfrac12k|\mathbf r_2-\mathbf r_1|^2$}
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.
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