Hooke's law from first principles

[Acut]

Following a discussion in this forum, I have a question: Is it possible to derive Hooke's law from first principles?

I think a purely electrostatic model is not adequate: Earnshaw theorem would imply there's no "relaxed" position. Also, electrostatic forces get weaker with increasing distance, the opposite trend of spring forces.

Is my reasoning correct? Have I overlooked something? Is it possible to create a purely electrostatic model of elasticity?

 

[Mapes]

Absolutely. In solids, atoms sit at an energy minimum (specifically, electrostatic attraction balanced by Pauli repulsion) that governs interatomic spacing. The energy values $E(x_0+\Delta x)$ around any minimum at $x_0$ can be expanded as a Taylor series,

$$E(x_0+\Delta x)=E(x_0)+\frac{\partial E(x_0)}{\partial x}\Delta x+\frac{1}{2}\frac{\partial^2 E(x_0)}{\partial x^2}(\Delta x)^2+\dots\approx \frac{1}{2}\frac{\partial E(x_0)}{\partial x}(\Delta x)^2$$

which is Hooke's Law where $k=\partial^2 E(x_0)/\partial^2 x$ (taking $E(x_0)$ as our energy reference and noting that $\partial E(x_0)/\partial x$ is zero because we're at an energy minimum). Does this answer your question?

 

[Acut]

You Taylor series reminded me we may model spring forces with nearly any kind of forces - since Hooke's law is linear, many forces will fit it for suficiently small displacements.

So there's something that avoids the consequences of Earnshaw's theorem- Pauli Repulsion.
Is Pauli Repulsion explained with quantum mechanics? Or is there a way of modelling this repulsion using only classical mechanics?

 

[Mapes]

Pauli repulsion is indeed explained with QM. And on the classical side, nuclear electrostatic repulsion may play a part too in controlling equilibrium atomic spacing, though I don't know offhand to what extent.

These effects are often modeled empirically with a repulsion term (e.g., in the http://en.wikipedia.org/wiki/Lennard-Jones_potential" ).