19/21 THE STATE-DETERMINED SYSTEM 



19/19. If a variable depends on velocity effects so that, for 

 instance 



1 ~f\dt' Xv x y 



dx 

 It 



dX 2 _ ft \ 



fa — J2\ X V X 2) 



dx 

 then if we substitute for -— inf^. . .) we get the canonical form 



dx 

 dx 



i/dt ==/i{/ 2 (a>i,0 a ), x v x 2 }^ 

 Jdt = f 2 (x 1 , x 2 ) ] 



19/20. If one variable changes either instantaneously or fast 

 enough to be so considered without serious error, then its value 

 can be given as a function of those of the other variables; and 

 it can therefore be eliminated from the system. 



19/21. Explicit solutions of the canonical equations 



dxjdt =fi(x v . . . , x n ) (i = 1, . . . , n) 



will seldom be needed in our discussion, but some methods will 

 be given as they will be required for the examples. 



(1) A simple symbolic solution, giving the first few terms of 

 x t as a power series in t, is given by 



e, = e**x\ (i = 1, . . . , n) . . (1) 



where X. is the operator 



fiK • • • ' O^o + • • • +/•«. ■ • ■ . «©5£ • ( 2 ) 



and e tx = 1 j rtX + t lx> +^X* + . . . . (3) 



It has the important property that any function 0(x v . . . , x n ) 

 can be shown as a function of t, if the x's start from x®, . . . , x„, 



by 0(x 19 . . . ,x n ) = e tx ${xl . . . , x») . . (4) 



(2) If the functions f t are linear so that 



dxjdt = a n x x -f a 12 x 2 -f . . . + a ln x n + b x 



dxjdt = a nl x x -f a n2 x 2 -f • • • + a nn x n + b n 

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(5) 



