On the symmetries of the modified Emden-type equation

TL;DR

This paper investigates the symmetries of the modified Emden-type equation derived from non-standard Lagrangians, showing that variational symmetries are contained within Lie symmetries, extending understanding of symmetry structures in such systems.

Contribution

It demonstrates that variational symmetries from non-standard Lagrangians are included in the Lie symmetries of the differential equation, revealing symmetry properties of the modified Emden-type equation.

Findings

Variational symmetries are embedded in Lie symmetries.

Non-standard Lagrangians lack kinetic and potential energy terms.

Symmetry analysis extends the understanding of modified Emden-type equations.

Abstract

For an autonomous system the symmetries of the Lagrangian are embedded in the symmetries of the differential equation. Recently, it has been found that the modified Emden-type equations follow from non-standard Lagrangian functions which involve neither the kinetic energy term nor the potential function. By working with one such Lagrangian we have calculated the Lagrangian symmetries and explicitly demonstrated that, as in the case of standard Lagrangian functions, the variational symmetries of the non-standard Lagrangian are also included in the Lie symmetries of the differential equation.