Group Theory in Physics

Group theory studies that has been considered by physicists as a very valuable tool for the clarification of the symmetry aspects of physical problems. The book intents to describe in detail the most important fundamental ideas of the group theory, its significant developments and various applications in: Hamiltonian systems, nonlinear systems, thermo-fluid dynamics, quantum mechanics and solid-state physics. In particular, different applications of Lie’s group theory to the above said fields are shown.The examination of the exact solutions of nonlinear equations takes an important place in physics. One of the noteworthy and efficient methods for gaining solutions of systems of nonlinear differential equations is the classical symmetries method, also called Lie’s group analysis. This method is employed for the constructions of solutions for the magnetohydrodynamic (MHD) flow of an upper-convected Maxwell (UCM) fluid over a porous shrinking wall, for the boundary layer equations for the Sisko fluid, and for a two-dimensional, unsteady flow and heat transfer of a viscous fluid over a surface in the presence of variable suction/injection. Another interesting application about the design of Lie’s group integrators of multibody system dynamics is presented. The quantum behavior of a physical system is a natural consequence of its symmetries. Hence, it is a fundamental to study the invariants of symmetry groups of them. In particular, invariant bilinear forms are very important for quantum physics, because these forms provide the link between mathematical description and experimental observations. The group theoretical analysis of the electronic and vibrational structure of the trimethine cyanine dye molecules is described. Other example of application of the group theory in the quantum mechanics is the establishment of a method for the description of an interacting spin-0 particle. The electronic energy band structure is a basic theory in condensed matter physics and can be used to study many physical properties of crystal materials. Here are presented a general method to unfold energy bands of supercell calculations to a primitive Brillouin zone and the results of the symmetry classification of the electron energy bands in graphene and silicene. The band degeneracy at high symmetry points or the existence of energy gaps, usually reflect the symmetry of the crystal, and this property is analyzed by considering two-dimensional (2D)-hexagonal lattices.