## 1. Solving a problem of optimum curves of descent using the enhanced Euler equation

 1. Solving a problem of optimum curves of descent using the enhanced Euler equation Horbulin V. P., Degtyarev A. V., 5. Shekhovtsov V. S. Yangel Yuzhnoye State Design Office, Dnipro, Ukraine; The National Academy of Sciences of Ukraine, Kyiv, Ukraine Kosm. teh. Raket. vooruž. 2020, (1); 3-12 https://doi.org/10.33136/stma2020.01.003 Language: Russian Annotation: The purpose of this study is the enhancement of Euler equation possibilities in order to solve the brachistochrone problem that is the determination of a curve of fastest descent. There are two circumstances: 1) the first integral of an Euler equation does not contain a partial derivative of integrand with respect to y in an explicit form; 2) when the classical Euler equation is derived, only the second term of integrand is integrated by parts. This allowed formulating a problem of determination of new conditions of functional extremality. It is assumed that the integrand of the first variation of a functional is equal to zero. Taking into account this pro vision and some other assumptions, the procedures have been determined for simultaneous application of the Euler equation and its analogue being non-invariant in relation to the coordinate system. The brachistochrone problem was solved using these equations: the curves that satisfy the conditions of weak minimum optimality were plotted. The time of a material point’s descent along the suggested curves and the classic extremals was numerically compared. It is shown that the application of suggested curves ensures short descent time as compared to the classic extremals. Key words: first variation of a functional, joint application of extremality conditions, non-invariance in relation to the coordinate system, parametric shape of the second variation, optimum curves of descent. Bibliography: 1. Bliss G. A. Lektsii po variatsionnomu ischisleniiu. М., 1960. 462 s. 2. Yang L. Lektsii po variatsionnomu ischisleniiu i teorii optimalnogo uravneniia. М.,1974. 488 s. 3. Elsgolts L. E. Differentsialnye uravneniia i variatsionnoe ischislenie. М., 1965. 420 s. 4. Teoriia optimalnykh aerodinamicheskikh form / pod red. А. Miele. М., 1969. 507 s. 5. Shekhovtsov V. S. O minimalnom aerodinamicheskom soprotivlenii tela vrashcheniia pri nulevom ugle ataki v giperzvukovom neviazkom potoke. Kosmicheskaia tekhnika. Raketnoe vooruzhenie: Sb. nauch.-tekhn. st. / GP "KB "Yuzhnoye". Dnipro, 2016. Vyp. 2. S. 3–8. 6. Sumbatov А. S. Zadacha o brakhistokhrone (klassifikatsiia obobshchenii i nekotorye poslednie resultaty). Trudy MFTI. 2017. T. 9, №3 (35). S. 66–75.