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X'LIX. Adiabatic Invariants of Mechanical Systems*. 

 By J. M. Burgers |. 



Introduction. 



IN the course o£ tlie past year the Theory of Quanta has 

 made great progress through the study o£ the so-called 

 "conditionally periodical" systems \. The characteristic 

 feature of these systems is that the integral of Action 



W = j2T.^ 



(T = kinetic energy) can be separated into a sum of inte- 

 grals, each of which depends on one only of the coordinates : 



W= 2 fojv^Jti (1) 



Generally the motion of each coordinate is a " Libration" : 

 it goes up and down between two fixed limits, the values 

 of which are determined by the integrated equations of 

 motion §. For these systems the principle which is used to 

 introduce the quantum of action li has the form 



h=2$dq k </Y k {q k j = n k .7i ... (2) 



[iik denotes an integral number) 

 (during the integration q k goes up and down once between 

 its limits). 



Now it has been shown by P. Ehrenfest || that for the 

 theory of quanta the '' adiabatic invariants " are of great 

 importance. These quantities, are functions belonging to 

 the system, which have the property that their value is not 

 changed when the system is d ; sturbed adiabatically (see 

 the precise definition by Ehrenfest, I. c. and below, § 1). 

 Especially he has shown that in the older forms of the 



* This article, Supplement No. 41e-41e of the Communications of the 

 Physical Laboratory at Leiden, appeared for the first time in the Proc. 

 Acad, of Amsterdam, xxv. (1916), pp. 849, 918, 1055. For those calcu- 

 lations which are treated only very shortly here the reader may be 

 referred to the original paper. 



+ Communicated by Prof. H. Kamerlingh Onnes. 



% K. Schwarzschild, Sitz.-Ber. Berl. Akad, 1916, p. 548. 



P. Epstein, Ann. d. PJn/s. vol. 1. (1916) p. 490; li. (1916) p. 168. 

 P. Pebye, Gbti. Nachr. 1916, p. 142; PMjs. Zeitschr. xvii. (19l6) 



pp. 507, 512. 

 A. Scmmerfeld, Phys. Zeitschr. xvii. (1916) p. 491. 

 A general class of these systems was considered for the first time by 

 P. Stackel (Comptes Rendus, cxvi. (1898) p. 485; cxxi. (1895) p. 489). 

 For an account of the theory, see C. L. Charlier, Die Mechanik d. Himmels y 

 i. Leipzig, 1902. 



§ Cf. Charlier, /. c. Compare also note * § 1 (p. 515). 

 II P. Ehrenfest, supra, p. 500. 



