Atoms consisted of negative light electrons and heavier positive nuclei, and their electric attraction fell with distance at the same rate as gravity, suggesting that electrons orbited in ellipses the way planets orbited the Sun. However, an additional effect was predicted: electro-magnetic processes would also make the electrons constantly lose energy by "broadcasting" it into space, like a miniature radio stations. Apparently, certain orbits were immune to such losses, and light was only emitted when an electron jumped from one to the other.
The simplest pattern of emitted colors was that of hydrogen, for which a remarkably accurate formula actually existed, discovered around 1885 by a Swiss high school teacher named Johannes Balmer. In 1914 the young Danish physicist Niels Bohr (he and his brother Harald, a mathematician, were the stars of Denmark's soccer team) discovered what seemed like an explanation for the formulas. Bohr showed that Balmer's formula was obtained naturally and accurately, if one assumed a new law of nature. By that law, electron orbits were stable if the "action variables" associated with their periodic motion were an integral multiple (i.e. 1,2,3... times) of a new physical constant, one previously known from other "quantum" effects on the atomic scale. Paul Ehrnfest proposed that that rule extended to other atoms, whose multiple electrons behaved like multiple planets.
At this point Albert Einstein called attention to the pendulum whose string was gradually shortened: its "adiabatic invariant", the product E times T, was almost constant. Could it be, he suggested, that any quantity that was adiabatically conserved in large-scale nature, was exactly conserved on the atomic scale?
That led to the early quantum theory of Sommerfeld, for hydrogen and hydrogen-like atoms. However, when Max Born tried it on helium (two electrons) his results disagreed with observed colors of helium light. The successful "wave mechanics" theory of Schroedinger, Heisenberg and Born, which in 1925-6 replaced Bohr's naive (and unexplained) principle, used a completely different approach.
Re-emergence of Adiabatic Invariance:
Adiabatic invariance again surfaced decades later, in the study of ions and electrons moving in space. As the story of Birkeland and Stoermer shows, this area held special interest to Scandinavian scientists seeking to understand the aurora. One of them was Hannes Alfvén (1970 Nobel prize) who in his 1950 book "Cosmical Electrodynamics" showed that for appropriate conditions a certain mathematical combination of the properties of ions and electrons was almost a constant.
A Useful Tool to understanding Plasmas
He apparently did not realize that this was an adiabatic invariant of the sort defined by Ehrenfest: this was pointed out at about the same time by the Russian physicists Lev Landau (Nobel, 1962) and Solomon Lifshitz, as a worked-out example for the student in their textbook on the theory of fields.
A "second" adiabatic invariant, also important in the theory of radiation trapped in the Earth's field, was derived by Grad, Longmire and Rosenbluth while studying the confinement of laboratory plasma, and a related "third" invariant was introduced shortly afterwards by Northrop and Teller.