Calculus of variations. Textbook (Q2701770)

The present book is devoted to an exposition of foundations of calculus of variations. The book was written on the basis of a course of lectures given by the first author at the Physics Department of the Novosibirsk State University and contains the basic results of classical calculus of variations.NEWLINENEWLINEThe book consists of fifteen sections and includes about 100 problems which successfully supplement a theoretical part and promote better understanding of presented results. Section 1 contains problems of classical calculus of variations. In particular, a) Bernoulli's problem on brachistochrone, b) the problem on a surface of rotation with the smallest area, and c) the problem on geodesic lines on a surface. Sections 2--4 are devoted to the simplest problem of calculus of variations on minimizing the integral \(I[y] = \int^x_{x_0} F(x,y(x),y'(x))\, dx\). It is shown that extremal functions are solutions of the Euler equation \(F_y - \frac{d}{dx}F_{y'} = 0\). The authors consider cases when the order of the equation can be reduced. In Sections 5 and 6 solving problems a) and b) from Section 1 are discussed. In Sections 7--10 the authors consider more general problems of calculus of variations. In particular, problems with several unknown functions and independent variables, variational problem with moving boundary. The Euler-Poisson equation and the Euler-Ostrogradski equation are deduced, a series of problems connected with the Ostrogradski-Hamilton variational principle are formulated. Sections 11 and 12 are devoted to isoperimetric problems. In Section 13 a variational problem on conditional extremum is considered and the method of Lagrange multipliers for its solution is discussed. Using the method of Lagrange multipliers the problem on geodesic lines on a surface is solved in Section 14. In Section 15 the authors discuss sufficient conditions for a local extremum for the simplest problem of calculus of variations. The book contains also some information about the history of calculus of variations. Thus, the appendix includes a biographic reference about I. Bernoulli and his famous letter ``Problema novum, ad cujus solutionem mathematici invitantur'', the first publication on calculus of variations.NEWLINENEWLINEThe book can be recommended to students familiar with the course of mathematical analysis.