Farooq M Umar, Naseem Anum, Wafo Soh C
Department of Basic Sciences & Humanities, College of E & ME, National University of Sciences and Technology (NUST), H-12, Islamabad, Pakistan.
Department of Mathematics, Faculty of Engineering and Computer Sciences, National University of Modern Languages (NUML), H-9, Islamabad, Pakistan.
Heliyon. 2023 Jun 9;9(6):e17059. doi: 10.1016/j.heliyon.2023.e17059. eCollection 2023 Jun.
In this work we use complex Lagrangian technique to obtain Noether-like operators and the associated conserved quantities of an Euler-Lagrange (EL) system. We show that the three new conserved quantities namely, Noether conserved quantity, Lie conserved quantity and Mei conserved quantity reported by Fang et al. [1] for an EL-system and even more in numbers by Nucci [2] can also be obtained via complex variational formalism. Generally, a linear system of EL-equations possesses maximum 8-dimensional algebra of Noether symmetries and Noether's theorem yields related 8-first integrals. However, our methodology produces 10 Noether-like operators and 10 corresponding invariant quantities for the underlying system of equations. Among those ten first integrals, three (as named above) are reminiscent to those found in [1]. In addition, from the remaining list of conserved quantities several are similar to those reported in [2]. Moreover, the current study presents an alternative approach to compute invariant quantities of EL-systems and leads to interesting and fascinating results.
在这项工作中，我们使用复拉格朗日技术来获得诺特定理类算子以及欧拉 - 拉格朗日（EL）系统的相关守恒量。我们表明，Fang等人[1]为一个EL系统报告的三个新的守恒量，即诺特守恒量、李守恒量和梅守恒量，以及Nucci[2]所发现的更多此类守恒量，也可以通过复变分形式得到。一般来说，一个EL方程的线性系统具有最大8维的诺特对称代数，并且诺特定理产生相关的8个第一积分。然而，我们的方法为基础方程组产生了10个诺特定理类算子和10个相应的不变量。在这十个第一积分中，有三个（如上所述）让人联想到[1]中所发现的那些。此外，从其余的守恒量列表中，有几个与[2]中报告的那些相似。而且，当前的研究提出了一种计算EL系统不变量的替代方法，并得出了有趣且引人入胜的结果。
