THE RAPIDLY DECREASING FUNCTIONS OF THE MICROSCOPICALLY-DESCRIPTIVE HYDRODYNAMIC EQUATIONS (Study of the History of Mathematics)
Shigeru Masuda · Kyoto University Research Information Repository (Kyoto University) · 2011
The two-constant' theory introduced first by Laplace in 1805 still forms the basis of current theory describing isotropic, linear elasticity, describing the capillarity.By using "two-constant" theory, the Navier-Stokes equations are formulated.These equations with the two coefficients in the ratio 1 : 3 originated from Poisson 16] in 1831.Moreover, these equations contained both a linear and a nonlinear term developed earlier in Navier $s$ equations $|20]$ in 1827.We show the process of formulation of calculus of vareations using the two functions characterized from the attraction and repulsion, and his criticism to Laplace imaging the Gaussian function as the rapidly decreasing function by Gauss in 1830.And we introduce a contribution to the hydromechanics, partly because he was a comtenporary of the epock of formulation of the Navier-Stokes equations, which are our main theme in our paper.Particularly, from the viewpoint of mathematics, several important topics such as integral theory in \S 4.3 which are his selling points.We show his unique rapidly decreasing function (we call it $RDF$ below) and reduction of integral from sextuplex to quadruplex, in the sections \S 4.1.In and after \S 4.2, we show his calculus of variations in the capillarity against the $RDF$ and calculation of it by Laplace.