The Noether–Bessel-Hagen theorem can be considered a natural extension of Noether Theorem to search for symmetries. Here, we develop the approach for dynamical systems introducing the basic foundations of the method. Specifically, we establish the Noether–Bessel-Hagen analysis of mechanical systems where external forces are present. In the second part of the paper, the approach is adopted to select symmetries for a given systems. In particular, we focus on the case of harmonic oscillator as a testbed for the theory, and on a cosmological system derived from scalar–tensor gravity with unknown scalar-field potential [Formula: see text]. We show that the shape of potential is selected by the presence of symmetries. The approach results particularly useful as soon as the Lagrangian of a given system is not immediately identifiable or it is not a Lagrangian system.

中文翻译：

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动力系统的 Noether-Bessel-Hagen 对称方法

Noether-Bessel-Hagen 定理可以被认为是 Noether 定理的自然延伸，以寻找对称性。在这里，我们开发了动态系统的方法，介绍了该方法的基本基础。具体来说，我们建立了存在外力的机械系统的 Noether-Bessel-Hagen 分析。在本文的第二部分，采用该方法为给定系统选择对称性。特别是，我们关注谐振子作为理论试验台的情况，以及从标量场势未知的标量-张量引力推导出的宇宙学系统[公式：见正文]。我们表明，势的形状是通过存在对称性来选择的。