**Earnshaw's Theorem**

A charged body cannot be held in stable equilibrium by electrostatic forces from other charged bodies.

"On the Nature of the Molecular Forces which Regulate the Constitution of the Luminiferous Ether,"

Extension of Earnshaw's theorem to containers with classical mobile charges at finite temperature:

Now consider the Casimir forces between a set of objects described by position and frequency dependent dielectric response:

"Constraints on Stable Equilibria with Fluctuation-Induced (Casimir) Forces,"

S.J. Rahi, M. Kardar, & T. Emig, Phys. Rev. Lett. 105, 070404 (2010)

This is valid also at finite temperatures.

Appears to rule out repulsion from metamaterials constructed out of ordinary metals and dielectrics.

Levitation can be achieved if conditions for the theorem are violated: