Earnshaw's Theorem, or why quantum mechanics was inevitable
In the later 19th Century the world of theoretical physics was to a large extent resting on its laurels. After big successes in the fields of gravitation and electromagnetism culminating in the work of Faraday and Maxwell, it was felt that the basic equations were OK and the only thing left to do was to solve them more and more accurately in different situations - a question of mathematics rather than fundamental physics. Even special relativity wasn't a big challenge to the classical picture, since it only meant that you had to exchange your Galilean transformations with Lorentz transformations. The nature of the stuff you apply the equations to -- matter itself -- was, however, a little more mysterious: it was, at least, generally accepted that it was made up of tiny constituents, which helped to explain the successes of physical chemistry and statistical mechanics.
With a little thought, people could have anticipated that a great revolution was inevitable at some point, since the known forces that could possibly act on the atoms were all inverse square forces. Gravity worked by Newton's law of gravitation, and forces between two electric or magnetic charges obeyed Coulomb's law. The force between two magnetic dipoles is not inverse square, but every dipole can be thought of as two monopole charges fixed together: you can get the correct dipole force law just by adding inverse squares together. (The rest of Maxwell's equations describe what happens when electric or magnetic charges move.)
But Earnshaw's Theorem, proved in 1839, guarantees that
there is no static stable configuration for point particles obeying an inverse square force law.
What does this mean? In order to make a chunk of matter, you need a setup of atoms or other constituents which is in stable equilibrium
: if one atom moves a little from its average position it should feel a force pushing it back. If not, matter would fall apart. And this is what the theorem tells us, there can't be a stable equilibrium state!
The case of gravity suggests one way out: instead of a static configuration, you could have a state of constant motion which leads to a stable periodic orbit. Maybe something like this is going on inside bits of matter, but on such a small scale that we can't detect it.
Impact on history
The discovery of the electron in 1897 by Thomson (which, by the way, hasn't been noded yet), using an apparatus strikingly similar to the device you are probably looking at right now, threw the problem into sharp relief: people realized that the atom had to contain charged constituent particles. Then a possible way for it to be stable was to have some of them orbiting round the others. This implies that the orbiting charges are accelerating toward the centre of rotation. But it was known (by solving Maxwell's equations) that if a charge is accelerating, it produces electromagnetic radiation--light--and loses energy. Electrons would spiral in inexorably and the atom would collapse in a minute fraction of a second, in a burst of radiation.
Thomson "fixed" the problem with his plum pudding model of the atom: electrons lived in a stable configuration inside a diffuse cloud or blob of positive charge. The blob kept them inside, and their mutual repulsion kept them from hitting one another. But how was the blob itself stabilized? There was no answer. The model was clearly incomplete. Besides, it would soon be disproved by Ernest Rutherford's discovery of the atomic nucleus (1911). This showed that most of the atom was empty space and its constituents were (to a good approximation) point particles with positive and negative charges. The Rutherford model of the atom was a good fit with experiment, but unfortunately completely inconsistent theoretically, since it had the problem of instability to swift collapse noted earlier.
Next, Niels Bohr stepped in with an brilliant fudge, based on borrowing a quantity from the Planck theory of radiation and Einstein's explanation of the photoelectric effect. Instead of being able to follow any path consistent with their equation of motion, the electrons were (don't ask how) restricted to have only a certain fixed amount of angular momentum, or multiples of this amount. At this point any respectable physicist would just throw up their hands in horror and wonder what rubbish was going to come next. Bohr was postulating that the normal rules of electromagnetism and dynamics should be ignored: despite having an acceleration, the electron simply could not radiate unless it made a sudden jump to a different value of the ang. mom., and when it reached the smallest value it just couldn't radiate at all.
Amazingly, the Bohr model of the atom -- despite being apparent nonsense -- gave the correct result for simple atoms. But the distaste of the respectable physicists was justified: it was the right answer for the wrong reasons. The real solution, when it came, was something even more unintuitive, although much more elegant: the Schrodinger equation, Pauli exclusion principle and fully-fledged quantum mechanics in which the electron doesn't have a definite position at all, just a wavefunction. Finally, with the correct apparatus for calculating, it was possible to see why the things that matter is made out of don't just have inverse square forces, but instead -- thanks to the properties of the constituents -- all sorts of other forces.
van der Waals forces are a prime example of what binds matter together, and although van der Waals brilliantly anticipated their existence many years before quantum theory (indeed, Isaac Newton, in his Opticks (1717), also suspected that some other type of force besides an inverse square was operating, and Rudjer Boskovic further developed the notion of interatomic forces ), a full explanation requires QM. Interestingly, the quantum mechanical picture of an atom is essentially the reverse of the Thomson plum pudding: the electron is delocalized in a bolb or cloud of negative charge in order to satisfy the Schrödinger equation, and the nucleus sits in the middle. (The nucleus has a wavefunction of its own, of course, but this is much less spread out because the mass is much larger.)
One consequence of Earnshaw's theorem for everyday life is that things don't levitate in thin air. Or at least, they can't do so without moving. (See also the Levitron.) However, as detailed at the end of my antigravity device writeup, quantum mechanical effects enable everyday objects to "levitate" quite satisfactorily, in the sense of maintaining their position with respect to other objects.
Back to the source
All of this illustrates what strange and violent things had to be done to the elegant structure of late 19th-century physics in order to evade Earnshaw's theorem and to have a chance of describing what stuff is made of. But enough of the quantum: what motivated Earnshaw to produce the theorem in the first place, at a time when no-one had the least idea about electrons and the rest of it? The answer is the luminiferous aether. Earnshaw investigated whether the aether could be a collection of particles held together by inverse square forces. Hence the title of his paper, "On the nature of the molecular forces which regulate the constitution of the luminiferous ether", Trans. Camb. Phil. Soc. 7, pp. 97-112 (1842). (Note that the paper wasn't published until three years after its presentation. Science was more relaxed in those days.)
And finally, the moment you've all been waiting for - the proof! It's actually quite simple, and like many other important things can be done in different ways.
The first thing is to establish the implications of a stable static equilibrium for the force acting on a single charged particle. Thanks to Newton's second law of motion, the force on a static particle must be zero. But in addition, if the particle is displaced by a small distance, the force must become nonzero and point back towards the original static position: it must be a restoring force. Hence, if we take a small spherical region of space centred on the static position, the force that would act on the particle if it were displaced to any point on the surface of the sphere must point inwards.
Now we can go several routes. The simplest is to take the description of inverse square forces by lines of force (which also isn't noded). A positive charge is a source of lines of force, which point from the positive to the negative charges which are the sinks. Lines of force never cross and cannot begin or end except on
charges. Gauss' law ensures that this will always work. The force on a particle placed in a given field pattern is in the direction of the lines and proportional to its charge. (Note that a particle doesn't feel the
lines of force it generates itself.) Let's suppose the particle is positively charged. (If it isn't, then replace positive by negative and inwards by outwards in what follows.)
Then, for a stable equilibrium, all the lines of force at the surface of the sphere have to be pointing inwards. But the only way for this to be true is if there is a negative charge inside the sphere. So let's shrink the sphere so that it's small enough to exclude any other charged particles apart from the one we're concerned with. This ends up with a reductio ad absurdum: either the configuration must be unstable or there must be a negative charge infinitely close to every positive charge, and vice versa: in other words the whole structure collapses to zero.
A slightly more mathematical proof uses Gauss' law directly: for a stable equilibrium, the normal surface integral of the force, and thus of the (electric, magnetic or gravitational) field, over the small spherical region must be negative, therefore there must be a negative charge living inside it, in contradiction to our implicit assumption that there is a nonzero distance between the particle in question and the rest of the structure. This also works for gravity, which (unlike electricity and magnetism) is only attractive: in this case the result implies that there must be a nonzero mass inside the small sphere in addition to the original particle.
Finally a yet more mathematical proof: the stability criterion implies that the divergence of the force on the particle should be nonzero and negative. However, if you work out the divergence of an inverse square force, it is zero at all points in vacuum (see also Maxwell's equations). Since the force acting on any given particle is the sum of inverse square forces generated by all the other particles, they can never produce a stable static equilibrium.