Chapter ' 



STABILITY 



5/1. The word "stability" is apt to occur frequently in discussions 

 of machines, but is not always used with precision. Bellman refers 

 to it as ". . . stability, that much overburdened word with an un- 

 stabilised definition". Since the ideas behind the word are of great 

 practical importance, we shall examine the subject with some care, 

 distinguishing the various types that occur. 



Today's terminology is unsatisfactory and confused; I shall not 

 attempt to establish a better. Rather I shall focus attention on the 

 actual facts to which the various words apply, so that the reader will 

 tend to think of the facts rather than the words. So far as the words 

 used are concerned, I shall try only to do no violence to established 

 usages, and to be consistent within the book. Each word used will 

 be carefully defined, and the defined meaning will be adhered to. 



5/2. Invariant. Through all the meanings runs the basic idea of 

 an "invariant" : that although the system is passing through a series 

 of changes, there is some aspect that is unchanging; so some state- 

 ment can be made that, in spite of the incessant changing, is true 

 unchangingly. Thus, if we take a cube that is resting on one face 

 and tilt it by 5 degrees and let it go, a whole series of changes of 

 position follow. A statement such as "its tilt is 1°" may be true at 

 one moment but it is false at the next. On the other hand, the 

 statement "its tilt does not exceed 6°" remains true permanently. 

 This truth is invariant for the system. Next consider a cone stood 

 on its point and released, like the cube, from a tilt of 5°. The state- 

 ment "its tilt does not exceed 6°" is soon falsified, and (if we exclude 

 reference to other subjects) so are the statements with wider limits. 

 This inability to put a bound to the system's states along some 

 trajectory corresponds to "instability". 



These are the basic ideas. To make them incapable of ambiguity 

 we must go back to first principles. 



5/3. State of equilibrium. The simplest case occurs when a state 

 and a transformation are so related that the transformation does 



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