10 



§ 2. Systems of one degree of freedom. 



As the simplest illustration of tlie principles discussed in the former section 

 we shall begin by considering systems of a single degree of freedom, in which case 

 it has been possible to establish a general theory of stationary states. This is 

 due to the fact, that the motion will be simply periodic, provided the 

 distance between the parts of the system will not increase infinitely with the time, 

 a case which for obvious reasons cannot represent a stationary state in the sense 

 defined above. On account of this, the discussion of the mechanical transfor- 

 mability of the stationary states can, as pointed out by Ehrenfest'), for systems of 

 one degree of freedom be based on a mechanical theorem about periodic systems due 

 to BoLTZMANN and originally apphed by this author in a discussion of the bearing of 

 mechanics on the explanation of the laws of thermodynamics. For the sake of the 

 considerations in the following sections it will be convenient here to give the proof 

 in a form which differs slightly from that given by Ehrenfest, and which takes 

 also regard to the modifications in the ordinary laws of mechanics claimed by the 

 theory of relativity. 



Consider for the sake of generality a conservative mechanical system of s 

 degrees of freedom, the motion of which is governed by Hamilton's equations: 



dt oqu at dp^ ^ ' ^ ^ ^ 



where E is the total energy considered as a function of the generalised positional 

 coordinates Çj, . . . qsanå the corresponding canonically conjugated momenta pj, . . ■ ps- 

 If the velocities are so small that the variation in the mass of the particles due to 

 their velocities can be neglected, the p's are defined in the usual way by 



dT 

 P'^W,.' (^= !'••■ *) 



where T is the kinetic energy of the system considered as a function of the 

 generalised velocities Qi, . . ■ qs iqk = -^1 and of q^, . . . qs- If the relativity modi- 

 fications are taken into account the jd's are defined by a similar set of expressions 

 in which the kinetic energy is replaced by T = 2' m„ c' ( 1 — l/l — u'-lc^), where 

 the summation is to be extended over all the particles of the system, and u is the 

 velocity of one of the particles and m^ its mass for zero velocity, while c is the 

 velocity of light. 



Let us now assume that the system performs a periodic motion with the 

 period a, and let us form the expression 



^, Pk (jk dt, 



(5) 



') P. Ehrenfest, loc, cit. Proc. Acad. Amsterdam, XVI, p. 591 (1914) 



