Teodor M. Atanackovic – författare

This book is devoted to the basic variational principles of mechanics: the Lagrange-D'Alembert differential variational principle and the Hamilton integral variational principle. These two variational principles form the main subject of contemporary analytical mechanics, and from them the whole colossal corpus of classical dynamics can be deductively derived as a part of physical theory. In recent years students and researchers of engineering and physics have begun to realize the utility of variational principles and the vast possi­ bilities that they offer, and have applied them as a powerful tool for the study of linear and nonlinear problems in conservative and nonconservative dynamical systems. The present book has evolved from a series of lectures to graduate stu­ dents and researchers in engineering given by the authors at the Depart­ ment of Mechanics at the University of Novi Sad Serbia, and numerous foreign universities. The objective of the authors has been to acquaint the reader with the wide possibilities to apply variational principles in numerous problems of contemporary analytical mechanics, for example, the Noether theory for finding conservation laws of conservative and nonconservative dynamical systems, application of the Hamilton-Jacobi method and the field method suitable for nonconservative dynamical systems,the variational approach to the modern optimal control theory, the application of variational methods to stability and determining the optimal shape in the elastic rod theory, among others.

This book is intended to be an introduction to elasticity theory. It is as­ sumed that the student, before reading this book, has had courses in me­ chanics (statics, dynamics) and strength of materials (mechanics of mate­ rials). It is written at a level for undergraduate and beginning graduate engineering students in mechanical, civil, or aerospace engineering. As a background in mathematics, readers are expected to have had courses in ad­ vanced calculus, linear algebra, and differential equations. Our experience in teaching elasticity theory to engineering students leads us to believe that the course must be problem-solving oriented. We believe that formulation and solution of the problems is at the heart of elasticity theory. 1 Of course orientation to problem-solving philosophy does not exclude the need to study fundamentals. By fundamentals we mean both mechanical concepts such as stress, deformation and strain, compatibility conditions, constitu­ tive relations, energy of deformation, and mathematical methods, such as partial differential equations, complex variable and variational methods, and numerical techniques. We are aware of many excellent books on elasticity, some of which are listed in the References. If we are to state what differentiates our book from other similar texts we could, besides the already stated problem-solving ori­ entation, list the following: study of deformations that are not necessarily small, selection of problems that we treat, and the use of Cartesian tensors only.

This book is devoted to the basic variational principles of mechanics: the Lagrange-D'Alembert differential variational principle and the Hamilton integral variational principle. These two variational principles form the main subject of contemporary analytical mechanics, and from them the whole colossal corpus of classical dynamics can be deductively derived as a part of physical theory. In recent years students and researchers of engineering and physics have begun to realize the utility of variational principles and the vast possi­ bilities that they offer, and have applied them as a powerful tool for the study of linear and nonlinear problems in conservative and nonconservative dynamical systems. The present book has evolved from a series of lectures to graduate stu­ dents and researchers in engineering given by the authors at the Depart­ ment of Mechanics at the University of Novi Sad Serbia, and numerous foreign universities. The objective of the authors has been to acquaint the reader with the wide possibilities to apply variational principles in numerous problems of contemporary analytical mechanics, for example, the Noether theory for finding conservation laws of conservative and nonconservative dynamical systems, application of the Hamilton-Jacobi method and the field method suitable for nonconservative dynamical systems,the variational approach to the modern optimal control theory, the application of variational methods to stability and determining the optimal shape in the elastic rod theory, among others.

This book is intended to be an introduction to elasticity theory. It is as­ sumed that the student, before reading this book, has had courses in me­ chanics (statics, dynamics) and strength of materials (mechanics of mate­ rials). It is written at a level for undergraduate and beginning graduate engineering students in mechanical, civil, or aerospace engineering. As a background in mathematics, readers are expected to have had courses in ad­ vanced calculus, linear algebra, and differential equations. Our experience in teaching elasticity theory to engineering students leads us to believe that the course must be problem-solving oriented. We believe that formulation and solution of the problems is at the heart of elasticity theory. 1 Of course orientation to problem-solving philosophy does not exclude the need to study fundamentals. By fundamentals we mean both mechanical concepts such as stress, deformation and strain, compatibility conditions, constitu­ tive relations, energy of deformation, and mathematical methods, such as partial differential equations, complex variable and variational methods, and numerical techniques. We are aware of many excellent books on elasticity, some of which are listed in the References. If we are to state what differentiates our book from other similar texts we could, besides the already stated problem-solving ori­ entation, list the following: study of deformations that are not necessarily small, selection of problems that we treat, and the use of Cartesian tensors only.

The books Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes and Fractional Calculus with Applications in Mechanics: Wave Propagation, Impact and Variational Principles contain various applications of fractional calculus to the fields of classical mechanics. Namely, the books study problems in fields such as viscoelasticity of fractional order, lateral vibrations of a rod of fractional order type, lateral vibrations of a rod positioned on fractional order viscoelastic foundations, diffusion-wave phenomena, heat conduction, wave propagation, forced oscillations of a body attached to a rod, impact and variational principles of a Hamiltonian type. The books will be useful for graduate students in mechanics and applied mathematics, as well as for researchers in these fields.Part 1 of this book presents an introduction to fractional calculus. Chapter 1 briefly gives definitions and notions that are needed later in the book and Chapter 2 presents definitions and some of the properties of fractional integrals and derivatives.Part 2 is the central part of the book. Chapter 3 presents the analysis of waves in fractional viscoelastic materials in infinite and finite spatial domains. In Chapter 4, the problem of oscillations of a translatory moving rigid body, attached to a heavy, or light viscoelastic rod of fractional order type, is studied in detail. In Chapter 5, the authors analyze a specific engineering problem of the impact of a viscoelastic rod against a rigid wall. Finally, in Chapter 6, some results for the optimization of a functional containing fractional derivatives of constant and variable order are presented.