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.

Annotation: The purpose of the article is development of a modified variational method to determine extremals in the tasks of search for optimal solutions. The method has been developed using the results of investigations of the first variation of functional with autonomous subintegral function for the problem with fixed ends. The assumption of non-zero values of variation of function at boundary points has been introduced. It is shown that when using this assumption and introducing some other assumptions and limitations, it is possible to expand the class of permissible functions, among which the extremal curves should be sought for. With this expansion, to construct one extremal it is necessary to use two conditions of extremeness, one of which is Euler equation. To fulfill them, it is necessary to realize the constancy of partial derivative from subintegral function of desired variable at each point of interval considered. The new condition of extremeness unlike Euler equation is noninvariant relative to coordinate system. The use of this property allows, at presentation of the second variation of functional in parametrical form, constructing the solutions that satisfy the necessary and sufficient conditions of local minimum (maximum). It is noted that the proposed method is the first step in the development of a new approach to solution of multidimensional variational problems. The use of the latter will allow obtaining new solutions of various problems of technical mechanics, such as the task of determining optimal trajectory parameters of launch vehicles in the phase of designing and development of technical proposals, selection of optimal flight modes et al. The efficiency of the proposed method is demonstrated by example of solving the known problem about brachistichrone – determination of the curve of quickest descent. Using the method, two curves have been constructed that satisfy the necessary and sufficient conditions of optimality. The results are presented of comparison of time of material point descent along the proposed curves and descent along classical extremals. It is shown that the time of descent along the proposed curves is shorter than that at descent along classical exteremals.

Annotation: To solve the problem, procedure is suggested forming the basis for two functional extremeness conditions, one of which is Euler equation. Application of this procedure enables to obtain set of extremals, each of them is defined by a value of one of constants. It is shown that with such an approach extremals, obtained for a definite range of the above constant, can give to functional stronger results than when applying a classical solution.