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Earnshaw's theorem states that the Laplacian of the potential energy of a small charge moving through a landscape full of static negative and/or positive charges (and gravity) is zero. Thus you can't have stable levitation (electrostatic or magnetostatic) because that would require a "bowl" in the potential energy field (levitation uses forces such as diamagnetism to circumvent the theorem and allow objects float).

What if the landscape charges are attached to springs and dampers so they move slightly when we move our charge? My intuition is that this would always create a negative (mountaintop) Laplacian. If there was a case which didn't it would be exploited for the magnetic levitation since diamagnetism is very weak. Consider an infinite sheet of plane charges. Without springs the force on our + point charge is constant. But with springs, the sheet will bulge away (if +) or toward (if -) our +. Both cases create a negative Laplacian. Is a negative Laplacian true for any spring configuration? If so, is there a simple way of proving it?

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Earnshaw's theorem relates to STATIC systems; ie no dynamic feedback control, and it explains why the EM force, which we can manipulate, cannot be used to compress atoms together to cause thermo-nuclear fusion.

The sun can create thermo-nuclear fusion, because gravity sucks, instead of pushes, so gravity can hold the material together dense enough and hot enough and long enough for fusion to occur. Electromagnetism can't do that, because of Earnshaw's theorem.

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