论文标题
大型散布的特性可在具有离散对称性的动态系统中振动。几何方面
Properties of large-amplitudes vibrations in dynamical systems with discrete symmetry. Geometrical aspects
论文作者
论文摘要
自上世纪后期90年代以来，罗斯托夫州立大学的研究小组一直在发展具有离散对称性的汉密尔顿系统中非线性正常模式（NNM）的灌木理论。开发了用于研究分子和晶体结构中的大振幅原子振动的组理论方法。每个灌木丛代表一定的振动模式集合，尽管这些模式的时间演变，但这些模式不会随时间变化，并且初始激发的能量仍然被困在灌木丛中。任何灌木丛的特征是其对称组，该组是系统对称组的子组。给定灌木中包含的模式由与对称性相关的方法确定，不取决于被考虑的系统中的原子间相互作用。该点和空间组的不可减至的表示基本用于NNM的灌木理论，并且该理论可以被视为大量振动情况下分子和晶体中众所周知的Wigner分类的概括性分类的概括。由于使用对称群的不可约说的表示可能是对布什理论初始熟悉的障碍，因此在本综述中，我们仅在普通的正常模式的帮助下解释了该理论的基本概念，这是从标准的教科书中众所周知的，考虑到机械系统中小原子振动的理论。我们的描述基于描述简单正方形分子的平面非线性原子振动的示例。
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.
