Michiyo Nakane, Kawasaki, Japan
Bohr’s Introduction to Action-Angle Variables in a 1918 Paper

Action-angle variables provide one of the most important mathematical techniques for quantum theory. These variables originated in Charlier’s books on celestial mechanics published in 1902 and 1907. Charlier performed a canonical transformation defined by a particular generating function and attained new canonical variables constructed by angle variables and action integrals. Noting Charlier’s argument, Schwarzschild defined action-angle variables and used them for an explanation of Stark effects in 1916. However, he did not present action variables in the form of I_i=∫p_idq_i, but he noted variables that have the same dimension as the action integrals that construct canonical variables together with angle variables, ω_i=n_it+β_i, where t is time, n_i is the mean motion, and β_i is the initial value of the angle. In a 1918 paper entitled “On the Quantum Theory of Line-Spectra,” Bohr mentioned that Kramers showed him a way to make I_i and ω_i canonical conjugates. An origin of formation of action-angle variables that we find in textbooks of mechanics today first appeared here. Then, Bohr developed his idea of a conditionally periodic system using the action variables I_i.