International Conference on Methods of Aerophysical Research, ICMAR 2008

The first variation problem of aerodynamics is the design problem of an axis-symmetrical forebody with given aspect ratio realizing a minimum of drag (Newton's problem – NP). It was solved by Newton [1] with using of the «local» formula (Newton's formula – NF) for the pressure р on a body’s surface. Later both mathematicians, and mechanics returned to NP, that stimulated the development of optimum control theory. The important element of the NP solution is the forward face, which has appeared as a boundary extremum segment (BES) because of the limitation of the body's length. In spite of the fact that NP was the first, or at least one of the first variation problem, the BES in calculus of variations classical courses, as a rule, were not considered. For this reason, the aerodynamicians that have approached to NP in 50-th years of the ХХ century and have knew only two-sided extremales, satisfying to Euler equation, not at once have managed to find the right solution or even to understand the Newton's outcomes. The questions with NP were arising either earlier. So, Legendre, allegedly, has given the recipe of improving the Newton's solution. The analysis of this recipe has shown, however, that the problem's formulation is needed to add the requirements of the NF applicability. As a result the forward face became the BES both by the length, and by the boundary of applicability the NF (A. Kraiko, 1963). The serious hardship have arisen when designing in NF frameworks of a fore-body, which is optimum for a given volume (instead of length). In these problem the new BES occur: a forward face of a given cylindrical body with the outstanding from it the sharp pin. In 60-70-th years of ХХ century G. Chernyi, A. Gonor, A. Miele, G. Saaris and D. Hull have solved the first variation problems about optimum spatial bodies in NF frameworks and also in NF with the local laws of friction. The drag of the fore-bodies, designed by them, with star-shaped cross sections has received noticeably less than that of the equivalent by the length and volume circular cones. Along with the local approaches based on NF and its generalizations, and also on the linear theory of two-dimensional supersonic flows, the approaches were developed from 1950 using equations of two-dimensional and axisymmetrical supersonic flow of ideal (inviscid and not heatconducting) gas (the Euler equations). Here the greatest promoting was reached in the design problem for a supersonic part of Laval nozzles realizing a maximum of thrust (K. Guderley, E. Hantsch, Yu. Shmyglevskii, L. Sternin, G.V.R. Rao, A. Kraiko, J. Armitage, V. Borisov, I. Michailov, V. Butov, I. Vasenin, A. Osipov, N. Tillyayeva) and after-bodies of a minimum wave drag. For fore-bodies streamlined with an attached shock wave, the partiсular solutions corresponding to (in the special cases) two-dimensional straight-line generatrix (G. Chernyi, 1950) and smooth contours of bodies of revolution with a duct (Yu. Shmyglevskii, 1957) were found. The solving of listed and other problems has required the development of body of mathematics that is of common interest for the optimum control theory of distributed parameter systems. It includes the method of a control contour – CCM (A. Nikolskii, 1950; K. Guderley and E. Hantsch, 1955); an uncertain control contour method – UCCM (A. Kraiko, 1979) as the validation of the G.V.R. Rao's approach (1958); a variation in a characteristic strip (G. Chernyi, 1950), a general Lagrange multipliers method – LMM (K. Guderley and J. Armitage, 1962); the allowing of the multipliers breaks along the characteristics in LMM (A. Kraiko, 1964); a variation of a focus of the centered waves of rarefaction (A. Kraiko, 1966). Having applied the LMM to designing of optimum