谐振子的辛欧拉分析方法

SYMPLECTIC EULER METHOD FOR HARMONIC OSCILLATOR

针对理想简谐振子力学模型,研究了其守恒律,并利用辛欧拉格式分析简谐振子振动过程.首先给出了谐振子系统的平方守恒律、周期守恒律和相差守恒律.构造了谐振子的普通欧拉格式和辛欧拉格式,研究了两种格式下三种守恒律各自的保持情况.模拟结果显示:辛欧拉格式能够精确保持时域守恒律(平方守恒律),但无法保持频域守恒律(周期守恒律和相差守恒律).如要克服辛欧拉格式的不足,需按邢誉峰教授提出的方法进行校正.

Focusing on the conservation properties, the symplectic Euler scheme of the harmonic oscillator was constructed to analyze its vibration properties. Firstly, three conservation laws, including the square conservation law, the period conservation law and the phase difference conservation law, were presented for the harmonic oscillator. And then, the common Euler scheme and the symplectic Euler scheme were constructed to study the above three conservation laws. The numerical results imply that the symplectic Euler scheme can preserve the conservation law in time domain （the square conservation law） exactly, but can＇t preserve the conservation laws in phase domain （the period conservation law and the phase difference conservation law） , which is the shortcoming of the symplectic method but can be overcome by the modification method presented by Prof. Xing.