Lieber 



The concept "Categories of Information" and the identification of the various 

 categories give us insight into the nature of applied mathematics and have pro- 

 duced the realization that applied mathematics is an art that uses the various 

 categories of information contrapuntally — where, for example, the same infor- 

 mation prescribed in more than one category of information does not in fact 

 constitute a redundancy, but is instead a viable instrument for further render- 

 ing explicit information otherwise restricted to the category of implicit infor- 

 mation. Specifically, introducing a simplifying assumption which, for example, 

 is appropriate to and correctly reports a fact about a particular aspect of fluid 

 flows, e.g., in the case of boundary layer theory, is tantamount to specifying 

 information in the category of explicit information, which the Navier- Stokes 

 equations purportedly already include in the category of implicit information. 

 As is well known, this procedure is neither redundant nor sterile. It is, in fact, 

 the very crux of applied mathematics and the only viable instrument which has 

 so far rendered explicit, and thereby useful, information which the Navier- 

 Stokes equations include in the category of implicit information. 



It is with this understanding that we originally formulated the principle of 

 minimum dissipation for viscous flows as a proposition which augments without 

 contradiction and/or redundancy the Navier- Stokes equations, and which in fact 

 rendered new and fundamental information about viscous flows which had not 

 been previously obtained by studies restricted to the Navier- Stokes equations 

 themselves. It was in this way that we first recognized that there exists a 

 linear substructure underlying the Navier-Stokes equations; that the prominence 

 of actual flows that tend to be potential over the principle part of a flow field, is 

 an aspect of the principle of minimum dissipation; that the principle of minimum 

 dissipation may be an aspect of a general stability principle according to which 

 a particular flow configuration among multiple configurations equally admitted 

 by the Navier-Stokes equations and boundary conditions is selected, thereby 

 suggesting a correspondence between hydrodynamic stability and minimum 

 dissipation. 



This in turn suggested to us a connection between a generalization of the 

 information first obtained as a theorem on the distribution of internal forces, 

 and a new general and fundamental law of mechanics. This new law includes 

 the propositions of classical mechanics as well as of a general stability princi- 

 ple which, in fact, gives expression to the evolutionary aspects and historical 

 thrust of the motivating forces in nature ~ aspects of force which the known laws 

 of classical mechanics do not express or include in any category of information. 



The realization that the principle of minimum dissipation gives only limited 

 expression to the principle of maximum uniformity as it was originally con- 

 ceived and formulated in terms of a positive, definite scalar measure of force, 

 prompted us to give it a more complete hydrodynamical expression by formu- 

 lating a new Hydrodynamical Variational Principle [6]. Although this varia- 

 tional principle does in part achieve this objective, it is nevertheless formulated 

 in terms of an integral of a function of gradients of energy, rather than directly, 

 in terms of a positive, definite, scalar force representation and measure of 

 non uniformity. 



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