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首页> 外文期刊>The Journal of the Astronautical Sciences >On Noether's Theorem and the Various Integrals of the Damped Linear Oscillator
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On Noether's Theorem and the Various Integrals of the Damped Linear Oscillator

机译：关于Noether定理和阻尼线性振荡器的各种积分

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摘要

Noether's theorem provides deep insight into the connection between analytical mechanics and the integrals of dynamic systems, specifically, showing how symmetries of the action integral are connected to the integrals of motion. To demonstrate Noether's theorem, the harmonic oscillator is often used as a simple example problem. Presentations in the literature, however, often focus on the single absolutely invariant symmetry for this problem. This paper presents a complete application of Noether's theorem to the damped harmonic oscillator, including general solutions of the divergence-invariant Killing equations and the associated integrals for all under damped, critically-damped, and overdamped cases. This treatment brings forward several interesting issues. Five different symmetries produce independent solutions to the Killing equations, but of course, only two independent integrals exist for this second-order system. Also, integrals of a particular desired form may not be produced directly from Noether's theorem and are referred to as non-Noether or asymmetric integrals. For the damped oscillator, one such example is the time-independent integrals, referred to as motion constants.
机译：Noether定理深入分析了分析力学与动态系统积分之间的联系，特别是说明了动作积分的对称性如何与运动积分相连。为了证明Noether定理，谐波振荡器通常用作一个简单的示例问题。但是，文献中的介绍通常集中于针对该问题的单一绝对不变对称性。本文介绍了Noether定理在阻尼谐波振荡器上的完整应用，包括发散不变的Killing方程的一般解以及在阻尼，临界阻尼和过阻尼情况下的所有积分。这种处理带来了几个有趣的问题。五个不同的对称性产生了Killing方程的独立解，但是对于该二阶系统，当然只有两个独立积分。同样，特定期望形式的积分可能不会直接从Noether定理产生，而是被称为非Noether或非对称积分。对于阻尼振荡器，这样的一个例子是与时间无关的积分，称为运动常数。

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