The history of the development of the theory of elliptic functions in the works of Abel, Jacobi, Weierstrass, Somov

The article explores the practical necessity of using elliptic functions. The history of the origin of the concept of an elliptic function is considered in detail. Clear conclusions on the formation of the apparatus of the theory of elliptic functions in the works of Abel, Jacobi, Weierstrass and Somov are proposed. Based on the proof of Abel's theorem, a representation of elliptic functions in terms of theta functions is shown. The introduction and use of elliptic and hyperelliptic functions bring the problems of control and orientation of mechanical objects to the simplest elements. The sought parameters of motion (direction cosines of the Euler angles) are the composition of such functions. General concepts and definitions of elliptic functions are reduced to the operation of integration. All methods of integration consist either in reducing the considered integral to elementary functions, or in its direct investigation, when this reduction is possible. Therefore, integral calculus is divided into separate sections. Among them, the first place after the theory of logarithmic and circular functions is occupied by the theory of elliptic functions. Giulio Carlo Fagnano (1682-1766, Italian mathematician, the first to pay attention to the theory of elliptic functions) discovered a remarkable relationship between arcs taken on one ellipse or one hyperbola. Euler proved analytically and generalized the property discovered by Fagnano. Soon John Landen (1719-1790, British mathematician, his transformations refer to elliptic integrals and elliptic functions) found that the arc of a hyperbola can be expressed in terms of two arcs belonging to ellipses with different eccentricities. The first systematic presentation on the theory of elliptic functions in Russia was presented by the St. Petersburg academician Osip Ivanovich Somov. This difficult and to this day branch of integral calculus is described in detail and clearly in his fundamental work "Foundations of the theory of elliptic functions" (1850). The book contains seven chapters, and a separate chapter is devoted to applications of elliptic functions to some questions of geometry and mechanics. In the presented article, the solution of the problem of the rotation of a rigid body about a fixed point, presented by Somov, will be presented.