19/9 THE STATE-DETERMINED SYSTEM 



(1) 



-^ -M*v • • • > *») 



w/^rtf iheps are single-valued, but not necessarily continuous, func- 

 tions of their arguments; in other words, the fluxions of the set 

 x ± , . . . , x n can be specified as functions of that set and of no 

 other functions of the time, explicit or implicit. The equations, in 

 this form, are said to be the canonical representation of the system. 

 (The equations will sometimes be written 



dxjdt =f(x 1 , . . . , x n ) (i = 1, . . . , n) . (2) 



and may be abbreviated even to x =f(x) if the context makes the 

 meaning clear.) 



(1) Let the system be state-determined. Start it at x\, . . . , x„ 

 at time t = and let it change to x v . . . , x n at time t, and then 

 on to x x + dx v . . . , x n + dx n at time t + dt. Also start it at 

 x v . . . , x n at time t = and let time dt elapse. By the group 

 property (S. 19/8) the final states must be the same. Using 

 the same notation as S. 19/8, and starting from x\, x t changes 

 to Fi (x°; t + dt and starting at x t it gets to F^x; dt). Therefore 



F^x ; t + dt) = Fix; dt) (i = 1, . . . , n). 



Expand by Taylor's theorem and write -xrF^a; b) as F'^a; b). 



Then 



F t .(*°; t) + dt.FfaO; t\ = Fi(x; 0) + dt.F&x; 0) 



(i = 1, . . . , n) 



But both F { {x°; t) and F { (x; 0) equal x t . 



. , n) . . (3) 

 . , n) 



dx 

 so by (3), -^ = F'i(x; 0) (i = 1, . . . , n) 



which proves the theorem, since Fi(x; 0) contains t only in 

 x v . . . , x n and not in any other form, either explicit or implicit. 



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