When discussing consequences of symmetries of dynamical systems based on Noether’s first theorem, most standard textbooks on classical or quantum mechanics present a conclusion stating that a global continuous Lie symmetry implies the existence of a time-independent conserved Noether charge which is the generator of the action on phase space of that symmetry, and which necessarily must as well commute with the Hamiltonian. However this need not be so, nor does that statement do justice to the complete scope and reach of Noether’s first theorem. Rather a much less restrictive statement applies, namely, that the corresponding Noether charge as an observable over phase space may in fact possess an explicit time dependency, and yet define a constant of the motion by having a commutator with the Hamiltonian which is nonvanishing, thus indeed defining a dynamical conserved quantity. Furthermore, and this certainly within the Hamiltonian formulation, the converse statement is valid as well, namely, that any dynamical constant of motion is necessarily the Noether charge of some symmetry leaving the system’s action invariant up to some total time derivative contribution. This contribution revisits these different points and their consequences, straightaway within the Hamiltonian formulation which is the most appropriate for such issues. Explicit illustrations are also provided through three general but simple enough classes of systems.

中文翻译：

---

Noether 对称性、运动的动力学常数和谱生成代数

在讨论基于诺特第一定理的动力系统对称性的后果时，大多数经典或量子力学标准教科书都提出了一个结论，即全局连续李对称意味着存在与时间无关的守恒诺特电荷，它是作用的产生者在该对称性的相空间上，并且必然也必须与哈密顿量对易。然而，这不必如此，该陈述也不能公正地说明诺特第一定理的完整范围和范围。更确切地说，适用的限制性要小得多，即相应的 Noether 电荷作为相空间上的可观察到实际上可能具有明确的时间依赖性，但通过具有不为零的哈密顿量的换向器来定义运动的常数，因此确实定义了一个动态守恒量。此外，这当然在哈密顿公式中，相反的陈述也是有效的，即任何运动的动态常数都必然是某种对称的诺特电荷，使系统的作用在某些总时间导数贡献中保持不变。本文直接在最适合此类问题的哈密顿公式中重新审视了这些不同的点及其后果。还通过三个通用但足够简单的系统类别提供了明确的说明。任何运动的动态常数必然是某种对称性的诺特电荷，使系统的作用在某些总时间导数贡献下保持不变。本文直接在最适合此类问题的哈密顿公式中重新审视了这些不同的点及其后果。还通过三个通用但足够简单的系统类别提供了明确的说明。任何运动的动态常数必然是某种对称性的诺特电荷，使系统的作用在某些总时间导数贡献下保持不变。本文直接在最适合此类问题的哈密顿公式中重新审视了这些不同的点及其后果。还通过三个通用但足够简单的系统类别提供了明确的说明。