ENERGY ACCELERATIONS, A STUDY IN ENERGY, &C. 



283 



initial mean kinetic energy is T^ -\- a and its initial rate of change is 

 zéro then on the li3q)otliesis that tlie coefficients reniain constant in the 

 équation of energy accélération we hâve at any succeeiliisg instant 



ri "1 .2 rd'-V'A 



^ ii>v~ \=^l\^-{- acosnl where //-^ = - -yr^ 



and the time average of tlie mean kinetic energy froni time U to 

 time / is 



a s in nt 

 = / y 



nt 



an expression initially ecjual to '1\ '\- a but Avhich soon approaches the 

 liniiting value 7'^ when the time-interval contains a considérable number 

 of periods. In this way the mean energy of the system tends towards its 

 equilibrium value, and the tendency is moreover unalfected by the rever- 

 sai of the initial velocity. In a system with one degree of freedom this is 

 as far we can get in reconciling irréversible transformations of energy 

 with the équations of motion of a réversible dynamical system, but 

 when there are a large number of degrees of freedom, and the corres- 

 ponding periods of energy oscillation are Inconnnensnrahle, it is clear 

 that the initial déviation from the equilibrium distribution will never 

 recur, and we may be sure that the energy at any instant will only 

 fluctuate slightly about its mean value. 



If the total energy of the particlc be given to be E the équation of 

 energy requires that 



This gives on multiplying by /(.r) dx Cp{v) dv and integrating 



[.V'"'. 



•]=|r^(.-)r/.. 



whence a necessary condition for energy equilibrium is 



r/.;/X^ 



E= 



l\dx 



)] 



d'^r 



'•^Idx-'^} 



