Properties of large-amplitudes vibrations in dynamical systems with discrete symmetry. Geometrical aspects

TL;DR

This paper reviews the theory of bushes of nonlinear normal modes in Hamiltonian systems with discrete symmetry, emphasizing geometrical aspects and using normal modes for explanation, with applications to atomic vibrations in molecules.

Contribution

It introduces a geometrical perspective on large-amplitude vibrations in symmetric systems and simplifies the bush theory using normal modes for better understanding.

Findings

Bushes trap energy in specific vibrational modes.

Symmetry groups determine the modes in a bush.

The theory generalizes Wigner's classification to large amplitudes.

Abstract

The research group from the Rostov State University has been developing the theory of bushes of nonlinear normal modes (NNMs) in Hamiltonian systems with discrete symmetry since the late 90s of the last century. Group-theoretical methods for studying large-amplitude atomic vibrations in molecular and crystal structures were developed. Each bush represents a certain collection of vibrational modes, which do not change in time despite the time evolution of these modes, and the energy of the initial excitation remains trapped in the bush. Any bush is characterized by its symmetry group, which is a subgroup of the system's symmetry group. The modes contained in the given bush are determined by symmetry-related methods and do not depend on the interatomic interactions in the considered system. The irreducible representations of the point and space groups are essentially used in the theory of the bushes of NNMs, and this theory can be considered as a generalization of the well-known Wigner classification of the small-amplitude vibrations in molecules and crystals for the case of large-amplitudes vibrations. Since using of the irreducible representations of the symmetry groups can be an obstacle to an initial familiarization with the bush theory, in the present review, we explain the basic concepts of this theory only with the aid of the ordinary normal modes, which is well known from the standard textbooks considering the theory of small atomic vibrations in mechanical systems. Our description is based on the example of describing plane nonlinear atomic vibrations of a simple square molecule.