Earnshaw s theorem

Cook R J 1979 Atomic motion in resonant radiation an application of Earnshaw s theorem Phys.Rev. 

As mentioned in Section 2.1, Earnshaw s theorem establishes that there can be no stable static equilibrium arrangement of classical ions and dipoles. Nevertheless, quantum mechanics allows numerous stable arrangements of ions, such as those... 

By Earnshaw s theorem, it is impossible to make a trap for ions static electric fields, but Penning in 1936 developed a stable ion trap that was a combination of a static uniform magnetic field and inhomogeneous electrostatic fields. In 1958 Paul and his associates made a successful electric quadrupole ion trap without a magnetic field by having the electric field oscillate sinusoidally at radio frequencies. 

This absence of effect, Lewis considered logically and scientifically objectionable for that state of motion which produces no physical effect whatsoever may better be called a state of rest. [Lewis, 1916]. And he was not alone in his scepticism. J. J. Thomson was trying to work a static atom theory as late as 1923 as were Langmuir, Davey, Born, Lande and Parson. All involved were trying to modify Coulomb s force law to get a theory in which the electrons in an atom remained still and were distributed at the corners of a cube. But in his Nobel lecture in 1922, Bohr delivered a sharp attack on static atom theories, pointing out that Earnshaw s theorem showed that a static distribution of charges must be unstable if Coulomb s law applies and that the argument could be extended to cover the proposed modified potentials. 

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