11 



which is easily seen to be independent of the special choice of coordinates q,, . . • '/.v 

 used to decribe the motion of the system. In fact, if the variation of the mass 

 with the velocity is neglected we get 



/ = 2\ TdL 



and if the relativity modifications are included, we get a quite analogous expres- 

 sion in which the kinetic energy is replaced by T" = >^-mf,v'''lVl ~ v^lc^. 



Consider next some new periodic motion of the system formed by a small 

 variation of the first motion, but which may need the presence of external forces 

 in order to be a mechanically possible motion. For the variation in / we get then 



It a s I s 



àl = \ ^{qk åpk + Pk Ô qk) (it -H : y^Pk qk â t 



^'o 1 1 1 



where the last term refers to the variation of the limit of the integral due to tlie 

 variation in the period tj. By partial integration of the second term in the bracket 

 under the integral we get next 



C" .." ' * 



0^^ = \ ^ iqkopk - pk<^qk)dt + I y^ Pk(qkàt + oqk) 



where the last term is seen to be zero, because the term in the bracket as well as 

 Pk will be the same in both limits, since the varied motion as well as the original 

 motion is assumed to be periodic. By means of equations (4) we get therefore 



ß<y s C'a 



i'n 1 Va 



Let us now assume that the small variation of the motion is produced by a 

 small external field established at a uniform rate during a time interval ä, long 

 compared with <t, so that the comparative increase during a period is very small. 

 In this case âË is at any moment equal to the total work done by the external 

 forces on the particles of the system since the beginning of the establishment of 

 the field. Let this moment he t ^ — S and let the potential of the external field 

 at t>0 be given by <J, expressed as a function of the q's. At any given moment 

 f > we have then 



dE = 



,-jO 



V » 



^2.ô^k"^'"~\2.8-qk^'''' 



si »'o 1 



which gives by partial integration 



