作用-角変数の形成におけるC.V.L. Charlierの寄与

抄録

Introduced by Karl Schwarzschild in 1916, action-angle variables provided effective mathematical tools with which to examine quantum phenomena. No historical work describes clearly how Schwarzschild came by the idea of them. This paper shows that the original idea of action-angle variables was substantially outlined in Charlier's two-volume textbook. Mechanics of Heaven, published in 1902 and 1907. In Volume 2. Charlier extended Jacobi's results and examined a leading function of a canonical transformation. He showed that a complete solution of the Hamilton-Jacobi equation of an intermediate orbit for given canonical equations becomes the leading function. The Hamilton-Jacobi equation for any intermediate orbit was found to be solvable. Charlier was then able to actually perform the canonical transformation, attaining new canonical variables that involved arbitrary constants of the solution to the equation of the intermediate orbits. He related the arbitrary constants to original canonical variables and changed the new canonical variables into new ones (ξ_i, η_i), that depend on an intermediate orbit. In this process, he used Stackers results as demonstrated in Volume 1. Charlier showed that if a Keplerian ellipsis is taken as the intermediate orbit. ξ_1 becomes an element of action integral multiplied by 1/π and η_i=η_it + β_i an argument of angle of a moving point, where n_i is frequency, t is time, and β_i is an arbitrary constant. Schwarzschild noted this fact and thereby attained his formal definition of action-angle variables.