DYNAMIC SYSTEMS 2/14 



variables and to the results of primary operations on them. It 

 is therefore a wholly objective property of the system. 



The concept of ' field ' will be used extensively for two reasons. 

 It defines the characteristic behaviour of the system, replacing the 

 vague concept of what a system ' does ' or how it 4 behaves ' 

 (often describable only in words) by the precise construct of a 



4 field '. From this precision comes the possibility of comparing 

 field with field, and therefore of comparing behaviour with 

 behaviour. The reader may at first find the method unusual. 

 Those who are familiar with the phase-space of mechanics will 

 have no difficulty, but other readers may find it helpful if at first, 

 whenever the word ' field ' occurs, they substitute for it some 

 phrase like ' typical way of behaving \ 



The Natural System 



2/14. In S. 2/4 a system was defined as any arbitrarily selected 

 set of variables. The right to arbitrary selection cannot be 

 waived, but the time has now come to recognise that both science 

 and common sense insist that if a system is to be studied with 

 profit its variables must have some naturalness of association. 

 But what is ' natural ' ? The problem has inevitably arisen 

 after the restriction of S. 2/3, where we repudiated all borrowed 

 knowledge. If we restrict our attention to the variables, we find 

 that as every real 4 machine ' provides an infinity of variables, 

 and as from them we can form another infinity of combinations, 

 we need some test to distinguish the natural system from the 

 arbitrary. 



One criterion will occur to the practical experimenter at once. 

 He knows that if an active and relevant variable is left unobserved 

 or uncontrolled the system's behaviour will become capricious, 

 not capable of being reproduced at will. This concept may 

 readily be made more precise. 



If, on repeatedly applying a primary operation to a system, it 

 is found that all the lines of behaviour which follow an initial state 



5 are equal, and if a similar equality occurs after every other initial 

 state S', S", . . . , then the system is regular. 



Whether a system is regular or not may be decided by first 

 constructing and then examining its field. For if the system is 

 regular, from each initial state will go only one line of behaviour, 



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