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 LII, A Theorem on the Dissipation of Energy, 



By S. H. BlTRBUEY*. 



I THINK the following theorem has never received a 

 formal proof. 



A material system being in motion, and there being dissi- 

 pation of energy by friction or otherwise, external forces, F, 

 are applied to maintain certain of the velocities constant. If 

 under these circumstances the system ultimately acquires a 

 motion in which it might move were there no dissipation and 

 no external forces, then in such motion the energy dissipated 

 per unit of time is minimum, given the constant velocities. 



For let #j, q 2 , . . . q n be the generalized coordinates defining 

 the position of the system; qi } q 2 ,...q n the corresponding 

 velocities; p x , p 2 , • • >p n the corresponding generalized compo^ 

 nents of momentum. 



Let q l} q 2 , . . . q,- be the velocities which are maintained 

 constant by the application of the forces F. 



Then we have for each coordinate, by Lagrange's equations, 

 dp_dT dU =F _dR 

 dt clq dq q dq ' 

 where T is the kinetic, U the potential energy, F g is the 

 generalized component of the external forces corresponding 

 to q, and H is Lord Rayleigh's dissipation function, namely 

 the energy dissipated per unit of time. The ultimate motion 

 is by hypothesis one in which the system might move were 

 there no dissipation and no forces F. That is ultimately, by 

 hypothesis, 



dt dq + dq ~°> 

 and 



for each coordinate. 



Now let us suppose that in the infinitely small time dt, the 

 given velocities q x . . . q r would be diminished by dissipation 

 alone by the quantities q\ . . . q' r . Then, in order to maintain 

 them constant, the forces F must in that same time dt produce 

 in the system the velocities q\. . . q' r . We may suppose the 

 forces Fdt applied impulsively at the end of the interval of time 

 dt to produce the velocities q\ . . . q f r . 



Now a set of impulses applied to produce in the system the 



prescribed velocities q\... q' r will generally produce at the 



same time other velocities q' r+l . . . q' n , the whole impulsive 



motion being, according to Thomson's theorem, determined 



* Communicated by the Author. 



