Symmetries and Lie Algebra of Ramanujan Equation

TL;DR

This paper performs symmetry analysis on Ramanujan's differential equations, deriving a related system and exploring its Lie algebra, revealing deep structural insights into the symmetries of these equations.

Contribution

It introduces a new related system of differential equations and analyzes its Lie algebra, linking it to the original Ramanujan system's symmetry structure.

Findings

The Lie algebra of the new system matches that of the original third-order equation.

The symmetry algebra contains a subalgebra representing the original system's symmetries.

The analysis suggests a general property of symmetries in systems derived from first-order ODEs.

Abstract

Symmetry analysis of Ramanujan's system of differential equations is performed by representing it as a third-order equation. A new system consisting of a second-order and a first-order equation is derived from Ramanujan's system. The Lie algebra of the new system is equivalent to the algebra of the third-order equation. This forms the basis of our intuition that for a system of first-order odes its infinite-dimensional algebra of symmetries contains a subalgebra which is a representation of the Lie algebra for any system or differential equation which can be obtained from the original system, even though the transformations are not point.