论文标题
在Noether定理与绝热不变理论之间的相互作用上
On the interplay between Noether's theorem and the theory of adiabatic invariants
论文作者
论文摘要
本文重点介绍将称为Rund-Trautman功能的重要数量。它已经在Noether定理中起着核心作用，因为它消失了对称性并导致保护法。本文的主要目的是展示在经典力学的领域中，伴随“几乎”对称性的“几乎”消失的rund-trautman函数会导致在绝热假设中的“几乎”运动常数，也就是说，即绝热不变。为此，首先详细介绍和分析了Rund-Trautman功能，然后将其用于一般的一维问题。最后，通过频率缓慢的谐波振荡器的示例来检查其在绝热环境中的相关性。值得注意的是，对于某些频率轮廓，通过它得出了绝热不变的显式扩展，并实现了说明性的数值测试。
This article focuses on an important quantity that will be called the Rund-Trautman function. It already plays a central role in Noether's theorem since its vanishing characterizes a symmetry and leads to a conservation law. The main aim of the paper is to show how, in the realm of classical mechanics, an 'almost' vanishing Rund-Trautman function accompanying an 'almost' symmetry leads to an 'almost' constant of motion within the adiabatic assumption, that is, to an adiabatic invariant. To this end, the Rund-Trautman function is first introduced and analysed in detail, then it is implemented for the general one-dimensional problem. Finally, its relevance in the adiabatic context is examined through the example of the harmonic oscillator with a slowly varying frequency. Notably, for some frequency profiles, explicit expansions of adiabatic invariants are derived through it and an illustrative numerical test is realized.
