关于谐振子第一积分的研究

A study on the first integrals of harmonic oscillators

提出一维谐振子基本积分的概念,并利用它们构造其他积分.将上述概念和方法推广到多维谐振子,利用直接构造法证明不同类型的二维谐振子都存在三个与时间无关的独立的第一积分,n维谐振子系统存在2n-1个与时间无关的独立的第一积分.讨论了分振动频率比为有理数和无理数的二维非对称谐振子的特征.

A concept of fundamental integrals of one-dimensional harmonic oscillator is presented, and other integrals can be constructed by use of fundamental integrals. The above concept and method are extended to multidimensional harmonic oscillators. By directly constructing other integrals from the fundamental integrals, it is proved that there are three independent time-independent integrals for all kinds of two-dimensional harmonic oscillators and there are 2n-1 independent time-independent integrals for n-dimensional harmonic oscillators. The characteristics of the anisotropic two-dimensional harmonic oscillator is discussed when the ratio between two frequencies is rational or irrational number.