论文标题
Noether定理的垂直扩展用于缩放对称性
Vertical extension of Noether Theorem for Scaling Symmetries
论文作者
论文摘要
本文的目的是提出一种新的方法来构建与动态系统缩放对称性相关的运动常数。缩放图可以是运动方程的对称性，而不是其相关的拉格朗日动作的对称性。我们在垂直扩展空间中构建了一个符合启发性的定理，该定理可用于获得这些对称性的运动常数。可以作为我们建筑的特定情况获得Noeth定理。为了说明该过程的工作原理，我们提出了两个有趣的示例，a）基于Schwarzian导数操作员的Schwarzian力学和b）Korteweg-de Vries（KDV）非线性部分微分方程在广义相对论的无效动力学的上下文中，广泛相对论是aDS $ _3 $ $ _3 $。我们还研究了Noether定理用于缩放对称性的倒数，并展示了我们如何构建和识别缩放转换的发生器，以及它如何用于我们能够构建的垂直运动常数。我们发现对与缩放对称性不是动作的对称性相关的对称性做出了有趣的贡献。最后，我们将结果与最近的分析和先前试图找到与这些美丽缩放定律相关的运动常数进行了对比。
The aim of this paper is to present a new approach to construct constants of motion associated with scaling symmetries of dynamical systems. Scaling maps could be symmetries of the equations of motion but not of its associated Lagrangian action. We have constructed a Noether inspired theorem in a vertical extended space that can be used to obtain constants of motion for these symmetries. Noether theorem can be obtained as a particular case of our construction. To illustrate how the procedure works, we present two interesting examples, a) the Schwarzian Mechanics based on Schwarzian derivative operator and b) the Korteweg-de Vries (KdV) non linear partial differential equation in the context of the asymptotic dynamics of General Relativity on AdS$_3$. We also study the inverse of Noether theorem for scaling symmetries and show how we can construct and identify the generator of the scaling transformation, and how it works for the vertical extended constant of motion that we are able to construct. We find an interesting contribution to the symmetry associated with the fact that the scaling symmetry is not a Noether symmetry of the action. Finally, we have contrasted our results with recent analysis and previous attempts to find constants of motion associated with these beautiful scaling laws.
