Editing Earnshaw's theorem (section)

==Explanation==

=== In electrostatics ===
Informally, the case of a point charge in an arbitrary static electric field is a simple consequence of [[Gauss's law]].  For a particle to be in a stable equilibrium, small perturbations ("pushes") on the particle in any direction should not break the equilibrium; the particle should "fall back" to its previous position.  This means that the force field lines around the particle's equilibrium position should all point inward, toward that position. If all of the surrounding field lines point toward the equilibrium point, then the [[divergence]] of the field at that point must be negative (i.e. that point acts as a sink).  However, Gauss's law says that the divergence of any possible electric force field is zero in free space. In mathematical notation, an electrical force {{math|'''F'''('''r''')}} deriving from a potential {{math|''U''('''r''')}} will always be divergenceless (satisfy [[Laplace's equation]]):

<math display="block">\nabla \cdot \mathbf{F} = \nabla \cdot (-\nabla U) = -\nabla^2 U = 0.</math>

Therefore, there are no local [[minimum|minima]] or [[maximum|maxima]] of the field potential in free space, only [[saddle point]]s. A stable equilibrium of the particle cannot exist and there must be an instability in some direction. This argument may not be sufficient if all the second derivatives of ''U'' are null''.''<ref>{{cite journal |last=Weinstock |first=Robert |date=1976 |title=On a fallacious proof of Earnshaw's theorem |journal=American Journal of Physics |volume=44 |issue=4 |pages=392–393 |doi=10.1119/1.10449|bibcode=1976AmJPh..44..392W }}</ref>

To be completely rigorous, strictly speaking, the existence of a stable point does not require that all neighbouring force vectors point exactly toward the stable point; the force vectors could spiral in toward the stable point, for example. One method for dealing with this invokes the fact that, in addition to the divergence, the [[curl (mathematics)|curl]] of any electric field in free space is also zero (in the absence of any magnetic currents).

=== In magnetostatics ===
It is also possible to prove this theorem directly from the force/energy equations for static [[magnetic dipole]]s (below). Intuitively, though, it is plausible that if the theorem holds for a single point charge then it would also hold for two opposite point charges connected together. In particular, it would hold in the limit where the distance between the charges is decreased to zero while maintaining the dipole moment – that is, it would hold for an [[electric dipole]]. But if the theorem holds for an electric dipole, then it will also hold for a magnetic dipole, since the (static) force/energy equations take the same form for both electric and magnetic dipoles.

As a practical consequence, this theorem also states that there is no possible static configuration of [[ferromagnet]]s that can stably [[Levitation (physics)|levitate]] an object against gravity, even when the magnetic forces are stronger than the gravitational forces.

Earnshaw's theorem has even been proven for the general case of extended bodies, and this is so even if they are flexible and conducting, provided they are not [[diamagnetic]],<ref>{{cite web |url=https://www.ru.nl/hfml/research/levitation/diamagnetic-levitation/levitation-possible/ |title=Levitation Possible |publisher=High Field Magnet Laboratory |author1=Gibbs, Philip |author2=Geim, Andre |access-date=2021-05-26 |url-status=live |archive-url=https://archive.today/20120908124057/http://www.ru.nl/hfml/research/levitation/diamagnetic/levitation_possible/ |archive-date=2012-09-08}}</ref><ref>{{cite journal|last=Earnshaw |first=S|author-link=Samuel Earnshaw|title=On the nature of the molecular forces which regulate the constitution of the luminferous ether|journal= Transactions of the Cambridge Philosophical Society|volume=7|pages=97–112| year=1842| url=https://archive.org/details/transactionsofca07camb/page/96/mode/2up}}</ref> as diamagnetism constitutes a (small) repulsive force, but no attraction.

There are, however, several exceptions to the rule's assumptions, which allow [[magnetic levitation]].

=== In gravitostatics ===
Earnshaw's theorem applies to static gravitational fields.

Earnshaw's theorem applies in an inertial reference frame. But it is sometimes more natural to work in a rotating reference frame that contains a fictitious [[centrifugal force]] that violates the assumptions of Earnshaw's theorem. Points that are stationary in a ''rotating'' reference frame (but moving in an inertial frame) can be absolutely stable or absolutely unstable. For example, in the [[Three-body problem#Restricted three-body problem|restricted three-body problem]], the effective potential from the fictitious centrifugal force allows the [[Lagrange point|Lagrange points]] L4 and L5 to lie at local maxima of the effective potential field even if there is only negligible mass at those locations. (Even though these Lagrange points lie at local maxima of the potential field rather than local minima, they are still absolutely stable in a certain parameter regime due to the fictitious velocity-dependent [[Coriolis force]], which is not captured by the scalar potential field.)