44 



In general it will not be possible to solve the equations of motion of the per- 

 liirbed system by means of separation of variables in a fixed set of positional coor- 

 dinates, but we shall see that the problem of the fixation of the stationary states 

 of the perturbed system may be attacked by a direct examination of the additional 

 energy of the system and its relation to the slow variations of the orbit, on the 

 basis of the usual theory of perturbations well known from celestial mechanics. 

 Consider a sj'stem for which every orbit, if undisturbed, is periodic independent of 

 the initial conditions, and let us assume that the equations of motion for some set 

 of coordinates q^, g,, ... Qs ai'e solved by means of the Hamilton-Jacobi partial 

 differential equation, given by formula (17) in Part I. The motion of the sj'stem is 

 then determined by the equations (18), and the orbit is characterised by means of 

 the constants «i,...rys, ß^, . . . ß^. If now the system is subject to some small 

 external field of force, the motion will no more be periodic, but, defining in the 

 usual way the osculating orbit at a given moment as the periodic orbit which 

 would result if the external forces vanished suddenly at this moment, we find that 

 the constants u^, ... as, ß^, . . . ßs, characterising the osculating orbit, will vary 

 slowly with the time. Assuming for the present that the external forces possess a 

 constant potential i? given as a function of the q's, we have according to the theory 

 of perturbations that the rates of variation of the orbital constants of the osculating 

 orbit will be given by') 



dak Sß dßk 8ß 



where ß is considered as a function of a^, . ■ 

 introducing for the q's their expressions as functions of these quantities obtained 

 by solving (18). The equations (44) allow to follow completely the perturbing 

 effect of the external field on the motion of the system. For the problem under 

 consideration, however, a detailed examination of the perturbations is not neces- 

 sary. In fact, we shall not be concerned with the small deformation of the orbit 

 characterised by the small oscillations of the orbital constants within a time inter- 

 val of the same order of magnitude as the period of the osculating orbit, but only 



(/c = 1, . 



. . s) (44) 



«s, ßl, ■ ■ 



. ßs and t, obtained by 



the discontinuitj' in qnestion would seem to be essentially connected with the form of the principles 

 of the quantum theory adopted in this paper. If for instance the quantum theory is taken in the 

 form pi-oposed by Planck in his second theory of temperature radiation, the consequent development 

 to periodic systems of several degrees of freedom would seem to involve a serious difficult}' as regards 

 the question of the necessary stability of the temperature equilibrium among a great number of systems 

 for small variations of the external conditions. In fact, in connection with the development of his theory 

 of the "physical structure of the phase space", mentioned in Part 1 on page 18, in which conditions 

 of the same type as (22J are established, Planck has deduced expressions for the total energy of a 

 great number of systems in temperature equilibrium, which, if applied to systems of the same kind 

 as those considered in the above example, show a dependency of this. energy on the temperature which 

 is different, according to whether polar coordinates or rectangular coordinates are used as basis for the 

 structure of the phase space. 



1) See f. inst. C. V. L Charlier, Die Mechanik des Himmels, Bd 1, Abt. 1, § 10. 



