Two-dimensional Systems that Arise from the Noether Classification of Lagrangians on the Line

TL;DR

This paper classifies Noether-like operators for 2D systems of ODEs using analytic continuation of Lagrangians, revealing new methods to find first integrals for nonlinear differential equations.

Contribution

It introduces a classification of Noether-like operators for 2D ODE systems via analytic continuation, expanding tools for constructing first integrals.

Findings

Maximal 8-dimensional Noether subalgebra for free particle equations

Identification of cases where Noether-like operators are also Noether symmetries

Effective methods for determining first integrals of nonlinear systems

Abstract

The Noether-like operators that play an essential role in writing down the invariants for systems of two ordinary differential equations (ODEs) are constructed. The classification of such operators is carried out with the help of analytic continuation of the Lagrangians on the line. Cases in which the Noether-like operators are also Noether symmetries for the systems of ODEs are briefly mentioned. In particular, the 8-dimensional maximal Noether subalgebra is remarkabely obtained for the simplest system of the free particle equations in two dimensions from the 5-dimensional complex Noether algebra. We present the effectivness of Noether-like operators as well as the determination of all first integrals of systems of nonlinear differential equation which have not been reported before. This study gives a new direction to construct first integrals for systems of nonlinear differential equations.