This article focuses on the recognition of 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 will be 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, especially within the adiabatic hypothesis for which the ‘almostness’ in question is in some sense measured by the slowness of time-dependent parameters. To this end, the Rund–Trautman function is first introduced and analyzed 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.

本文着重于识别一个重要的量，称为 Rund-Trautman 函数。它已经在诺特定理中发挥了核心作用，因为它的消失表征了对称性并导致了守恒定律。本文的主要目的是展示在经典力学领域中，伴随“几乎”对称性的“几乎”消失的朗德-特劳特曼函数如何导致“几乎”运动常数，尤其是在绝热假设中在某种意义上，所讨论的“几乎”是通过时间相关参数的缓慢性来衡量的。为此，首先对Rund-Trautman函数进行了详细的介绍和分析，然后针对一般的一维问题进行了实现。最后，它在绝热环境中的相关性通过具有缓慢变化频率的谐振子的示例进行检查。值得注意的是，对于某些频率分布，通过它导出了绝热不变量的显式扩展，并实现了说明性的数值测试。