Conserved quantities of Euler-Lagrange system via complex Lagrangian

Abstract

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.

Keywords: Conserved quantity; Euler-Lagrange system; Noether-like operator.