Lie be r 



natural phenomena — it is a statement of a natural law. A variational method is 

 a mathematical scheme for deriving the analytical consequences from a state- 

 ment that expresses the stationary quality of a certain functional, and in particu- 

 lar of a variational principle which is so formulated. In the case noted above, 

 the information content of the variational principle which we formulated is es- 

 sentially equivalent to that of the Navier- Stokes equations, which are its Euler- 

 Lagrange equations. The Navier- Stokes equations express the proposition that 

 all forces acting on each and every element of the materials in nature which are 

 adequately modeled by a Newtonian fluid, are constantly in equilibrium, i.e., 

 everjrwhere and for all time. The statement which formally defines a Newtonian 

 fluid expresses a connection between force geometry and time, and is of the 

 nature of a force law which is however restricted to, and thus identifies, cer- 

 tain macromechanical features of a class of natural materials. The known laws 

 of mechanics which are statements of the equilibrium of forces, are not laws of 

 force, but are instead propositions asserting certain necessary and constant 

 connections that are maintained between all forces acting on any and every com- 

 ponent of a classical mechanical system. 



The particular simple connection between forces, by which mechanical 

 equilibrium is defined, is a particular aspect of uniformity which according to 

 the laws of classical mechanics is constantly maintained everywhere in space 

 and always in time, for all material bodies. The fact that the known laws of 

 mechanics do not determine the so-called motivating forces which are included 

 in their formulation and which express a connection between them and the mo- 

 tion of material bodies, is made abundantly clear when we consider a substance 

 continuously extended in space as a model for depicting the macromechanical 

 characteristics of systems consisting of a very large number of discrete bodies. 

 In so doing, we obtain, by applying the laws of mechanics, a set of three scalar 

 equations which conditions the three components of acceleration, and the space 

 derivatives of nine components of the stress tensor of each element of the ma- 

 terial continuum. The resultant force externally applied to a characteristic 

 element of such a material continuum depends on the stress tensor by which it 

 is externally joined to the universe in which it is situated. By applying the laws 

 of classical mechanics to such a model, we obtain a set of three scalar differen- 

 tial equations which relate the three components of acceleration of a character- 

 istic element of the material, to the partial coordinate derivatives of the nine 

 components of stress, which designate the resultant force by which such a char- 

 acteristic element of the continuously extended material is externally joined to 

 the universe. From a strictly mathematical viewpoint it is obvious that the nine 

 components of stress in terms of which these forces are formally written are 

 in principle not determined by the principles of mechanics. Indeed, from a 

 mathematical point of view these stresses are highly undetermined , even if we 

 ascribe very strong properties of continuity and differentiability to the material 

 substratum. It is for this reason that it has been necessary to specify constitu- 

 tive relations between stress, geometry, and time in order to obtain an equiva- 

 lence between the number of relations and the number of parameters used to 

 describe the mechanical features of the system. 



The constitutive relations are tantamount to restricted force laws by which 

 the macromechanical properties of particular classes of materials are charac- 

 terized. Augmenting the laws of classical mechanics by general and/or restricted 



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