34, 35] HAMILTON'S PRINCIPLE. 99 



the difference of kinetic and potential energies is a minimum for the 

 actual path between given configurations as compared with infinitely 

 near paths which might be described (for instance under constraints) 

 in the same time between the same configurations; or more freely: 

 Nature tends to equalize the mean potential and kinetic energies 

 during a motion. 



Hamilton's principle is broader than the principle of energy, 

 inasmuch as U may contain the time as well as the coordinates. It 

 is true even for non- conservative systems (where a force -function U 

 does not exist or where U contains the time), if we write instead 

 of dU, 



We have then 



4) {dT + Z(Xdx + Y8y + Zde)}dt = 0. 



35. Principle of Least Action. It is to be noted that in 

 the statement of Hamilton's principle the infinitely near motion with 

 which the actual motion is compared is perfectly arbitrary (except 

 that it satisfies the equations of condition) ; so that to make the 

 system actually move according to the supposed varied motion might 

 require work to be done upon it by other forces. The paths described 

 by the various particles are not necessarily geometrically different 



It is proved in the integral calculus that the manner of subdivision into the 

 intervals <? s is immaterial. We may accordingly put them all equal so tha*t 



<y = ~ a > then dividing by (b a) we have 



that is the definite integral of a function in a given interval divided by the 

 magnitude of the interval represents the limit of the arithmetical mean of all 

 the values of the function taken at equidistant values of that variable throughout 

 the interval when the number of values taken is increased indefinitely. The 

 specification of the variable with respect to which the values are equally 

 distributed is of the first importance. For instance suppose that we change to 

 a new variable such that x = y(y], y = y-i(x) then 



6 y 



jf(x)dx = lf(x)q> 



The integral may now be interpreted as the mean of the function f(x)<p'(y) 

 multiplied by the interval through which y varies, for equally distributed values 

 of y. Thus we deal above with time means and space means. 



