"In this book, an introduction to the calculus of variations is presented and some applications to analytical mechanics are pointed out. In the last part of the book, a lot of problems and exercises are proposed and detailed solutions of them are given, which makes this book suitable for students in the latter years of a mathematics and physics degree.
The geometric methods used in the calculus of variations are presented in the first part of the book, 'In troduction to the Calculus of Variations'. Chapter 1, 'Elementary Methods to the Calculus of Variations', starts with the presentation of the free extremum problem and the constrained extremum problem. Then, methods to find extremum of a free functional and for a constrained
functional are deduced. Moreover, the Euler equations are derived. Particularly, the Euler equations become Lagrange equations via the Hamilton principle. The chapter ends with an application, namely the geodesics of a surface.
In Chapter 2, 'Variation of Curvilinear Integrals', the methods described in the previous chapter are extended to the calculation of the variation of curvilinear integrals. The deformation vector field of a curve associated with a family of curves is introduced. Then the variation of the circulation of a vector field is computed. Two applications are discussed: the optical path followed by light in a medium with a variable refractive index and the problem of isoperimeters.
In Chapter 3, 'The Noether Theorem', the notions of Lie group and invariant integral under a Lie group are introduced. In addition, the Noether theorem is proved. As example, Fermat s principle leads to a first integral for the above-mentioned application in optics.
Part 2 of the book, 'Applications to Analytical Mechanics', starts with Chapter 4, 'The methods of Analytical Mechanics'. The concepts of virtual displacement, holonomic constraint, and non-holonomic constraint are presented and the Lagrange equations in the general case of holonomic and non-holonomic constraints are deduced, via d Alembert s principle. In addition, the linearized motion equation of vi brating string is obtained and studied. Then the Hamilton equations of motion are presented. Using the Maupertuis principle, two applications are considered, namely the case of a material point and introduc tion to the Riemannian geometry. In Chapter 5, 'Jacobi s Integration Method', the integration methods for equations in mechanics are analyzed using Jacobi method and its application in the important case of Liouville s integrability. In the case of the motion of a material point with mass in the three-dimensional space, the generating Jacobi function is obtained in the cases of Cartesian coordinates, cylindrical representation, and spherical representation. The action-angle variables are introduced. As example, the three-dimensional harmonic oscillator is considered. Part 2 ends with Chapter 6, 'Spaces of Mechanics-Poisson Brackets'. Different equivalent configurations of spaces in analytical mechanics are pointed out. The concepts of symplectic scalar product of vectors, dynamical variable, Poisson bracket, first integral are introduced. Some properties of the Poisson bracket are discussed." --zbMath, 2020, Cristian Lazureanu, reviewer, expert opinion
