Group representations in Lagrangian mechanics: An application to a two-dimensional lattice

Abstract

Computational aspects of the representation theory of finite Abelian groups provide means for exploiting the symmetries of classical systems of harmonic oscillators. Coupled one-dimensional lattices and the two-dimensional rhombic crystal are modelled as systems of arbitrarily many point masses interconnected by ideal springs. After application of the Born cyclic condition, the Lagrangian functions for these systems are written in matrix notation. The irreducible matrix representations of the symmetry groups for these lattices generate unitary transformations into symmetry coordinates. Under these transformations, the Lagrangian matrices are either diagonalized or reduced to block diagonal form, thereby separating the equations of motion to the maximum extent made possible by system-wide considerations of a purely geometrical nature. Natural lattice frequencies are computed once the transformations to symmetry coordinates have been made.