, ._.■ , L/ieber 



dimensional Euclidean manifold in which the motion and state of a classical me- 

 chanical system consisting of N bodies free of prescribed forces, are described 

 and represented. Hertz restricts the admissible motions and states of the me- 

 chanical system, by formally subjecting the coordinates of its bodies to con- 

 straints which in the most general case are considered as nonintegrable and 

 therefore nonholonomic. As the application of these geometrical constraints to 

 a body restricts its freedom geometrically, these restrictions must emerge in 

 the Newtonian scheme as forces. 



The study based on the Gauss-Hertz formulations* has produced two results 

 which bear on the conception of the principle of maximum uniformity and on the 

 identification of a physical, i.e., of an ontological-geometrical basis for the pro- 

 duction of actual, stringent, holonomic as well as nonholonomic constraints in 

 nature's space-time manifold. This ontological-geometrical basis gives physi- 

 cal support and justification for the existence in nature of the geometrical re- 

 strictions which Hertz used to effect a formal reduction of force to geometry, 

 and serves to identify the formal representations he gave to nonholonomic con- 

 straints, with experience and thus with nature. 



The same study revealed that Hertz's construction, in which he formulated 

 a general law governing the motion of forceless mechanical systems subjected 

 to nonholonomic geometrical constraints, and which he showed renders valid all 

 previously known formulations of the laws of classical mechanics, also accom- 

 modates the formulation of a new and general stability law cited above. This 

 law which bears the same kind of general relation to stability as the established 

 laws of classical mechanics do to equilibrium, is found to be independent of the 

 known laws of mechanics and to embrace fundamental and general information 

 not included in these laws. This information bears on historical thrust and 

 commitment and derives from an adaptive -evolutionary process ascribed di- 

 rectly to the geometrical restrictions which impress nonuniformities on the 

 space-time manifold, and from which all forces are understood here to emerge. 

 This entails the identification and classification of holonomic and nonholonomic 

 ontological-geometrical constraints into the following types: (a) Active Strin- 

 gent Constraints, (b) Passive Stringent Constraints, and (c) Conditionally Strin- 

 gent Passive Constraints. This classification led naturally to the idea that the 

 annihilation of conditionally stringent passive constraints which are ascribed 

 here to universal congruence restrictions impressed on the space-time mani- 

 fold by the irreducible universals identified by the dimensional universal con- 

 stants, constitutes a fundamental and general instrument of adaptation in the 

 space -time manifold. It is this crucial instrument that allows one to conceive 

 and posit a general stability law for classical mechanical systems and that af- 

 fords, according to the principle of maximum uniformity, the mechanism which 

 is essential for physically producing the required many -to-one mappings evident 

 in biological systems. 



The annihilation of conditionally stringent constraints is accompanied by 

 consequent modifications of the forces emanating from the nonuniformities in- 

 duced by them in nature's space -time structure. According to the observations 



*Some results of this study are presented in Refs. 1 and 2. 



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