Similarity and Symmetry Methods: Applications in Elasticity and Mechanics of Materials (Lecture Notes in Applied and Computational Mechanics Book 73)

The principle aim of the book is to present a self-contained, modern account of similarity and symmetry methods, which are important mathematical tools for both physicists, engineers and applied mathematicians. The idea is to provide a balanced presentation of the mathematical techniques and applications of symmetry methods in mathematics, physics and engineering. That is why it includes recent developments and many examples in finding systematically conservation laws, local and nonlocal symmetries for ordinary and partial differential equations. The role of continuous symmetries in classical and quantum field theories is exposed at a technical level accessible even for non specialists. The importance of symmetries in continuum mechanics and mechanics of materials is highlighted through recent developments, such as the construction of constitutive models for various materials combining Lie symmetries with experimental data. As a whole this book is a unique collection of contributions from experts in the field, including specialists in the mathematical treatment of symmetries, researchers using symmetries from a fundamental, applied or numerical viewpoint. The book is a fascinating overview of symmetry methods aimed for graduate students in physics, mathematics and engineering, as well as researchers either willing to enter in the field or to capture recent developments and applications of symmetry methods in different scientific fields. Front Matter....Pages i-vii Some Recent Developments in Finding Systematically Conservation Laws and Nonlocal Symmetries for Partial Differential Equations....Pages 1-59 Construction of Conservation Laws Using Symmetries....Pages 61-164 Symbolic Computation of Nonlocal Symmetries and Nonlocal Conservation Laws of Partial Differential Equations Using the GeM Package for Maple....Pages 165-184 Symmetries of Hamiltonian Systems on Symplectic and Poisson Manifolds....Pages 185-269 Symmetries in Mechanics: From Field Theories to Master Responses in the Constitutive Modeling of Materials....Pages 271-351 Symmetries and Some Special Solutions of the Helfrich Model....Pages 353-364 Lie Group Analysis of the Willmore and Membrane Shape Equations....Pages 365-376