2/16 DYNAMIC SYSTEMS 



y = 2-1. On line 2 the state x = 0, y = 2-0 occurred again; but 

 after 0-1 seconds the state became x = 0-1, */ = 1-8 and not 

 x = 0-2, y = 2-1. As the two states that follow the state x = 0, 

 y = 2-0 are not equal, the system is not state-determined. 



A well-known example of a state-determined system is given 

 by the simple pendulum swinging in a vertical plane. It is known 

 that the two variables — (x) angle of deviation of the string from 

 vertical, (y) angular velocity (or momentum) of the bob — are 

 such that, all else being kept constant, their two values at a 

 given instant are sufficient to determine the subsequent changes 

 of the two variables (Figure 2/15/1). 



The field of a state-determined system has a characteristic 

 property: through no point does more 

 than one line of behaviour run. This 

 fact may be contrasted with that of a 

 system that is not state-determined. 

 Figure 2/15/2 shows such a field (the 

 system is described in S. 19/13). The 

 system's regularity would be established 

 if we found that the system, started at 



A, always went to A', and, started at 



B, always went to B' . But such a 

 system is not state-determined; for to 

 say that the representative point is 

 leaving C is insufficient to define its 



future line of behaviour, which may go to A' or B '. Even if the 

 lines from A and B always ran to A' and B', the regularity in no 

 way restricts what would happen if the system were started at 

 C: it might go to D. If the system were state-determined, the 

 lines CA', CB\ and CD would coincide. 



Figure 2/15/2 : The field 

 of the system shown in 

 Figure 19/13/1. 



2/16. We can now return to the question of what we mean when 

 we say that a system's variables have a ' natural ' association. 

 What we need is not a verbal explanation but a definition, which 

 must have these properties: 



(1) it must be in the form of a test, separating all systems into 



two classes; 



(2) its application must be wholly objective; 



(3) its result must agree with common sense in typical and 



undisputed cases. 



27 



