Low Dimensional Vessiot-Guldberg-Lie Algebras of Second-Order Ordinary Differential Equations

Rutwig Campoamor-Stursberg

doi:10.3390/sym8030015

Abstract

A direct approach to non-linear second-order ordinary differential equations admitting a superposition principle is developed by means of Vessiot-Guldberg-Lie algebras of a dimension not exceeding three. This procedure allows us to describe generic types of second-order ordinary differential equations subjected to some constraints and admitting a given Lie algebra as Vessiot-Guldberg-Lie algebra. In particular, well-known types, such as the Milne-Pinney or Kummer-Schwarz equations, are recovered as special cases of this classification. The analogous problem for systems of second-order differential equations in the real plane is considered for a special case that enlarges the generalized Ermakov systems.

Highlights

The theory of Lie systems, i.e., systems of non-autonomous first-order ordinary differential equations admitting a superposition principle, is an old one and emerges principally from the pioneering work of Lie, Vessiot and Guldberg in the late 19th century [1]
For each of the Lie algebras g listed in Proposition 1, we enumerate the realization by vector fields, the possible constraints on the functions in their components and the generic SODE Lie system admitting g as a Vessiot-Guldberg Lie algebra
By means of a direct approach, scalar SODE Lie systems admitting a Vessiot–Guldberg–Lie algebra of a dimension at most three have been constructed. These equations encompass some of the well-known types of SODE Lie systems, like the Liouville-type equations and the Milne–Pinney and Kummer–Schwarz equations

Summary

Introduction

The theory of Lie systems, i.e., systems of non-autonomous first-order ordinary differential equations admitting a (generally nonlinear) superposition principle, is an old one and emerges principally from the pioneering work of Lie, Vessiot and Guldberg in the late 19th century [1]. Proposition 1: If the vector fields Equation (15) generates a three-dimensional Vessiot–Guldberg–Lie algebra LVG , it belongs to one of the following types: 1. Equation (15), as well as the differential equations admitting g as Vessiot–Guldberg–Lie algebra LVG are obtained by solving the successive conditions imposed by the commutators of g. It merely remains to consider the condition specified by the component in ∂v of [ X1 , X3 ], given by: dk d2 G This constraint equation shows that the realization Equations (24) and (25) depends on an arbitrary function G ( x ). For any pair of functions { G ( x ) , k ( x )} satisfying Equation (26), the commutators Equation (18) is satisfied and defines a three-dimensional Vessiot–Guldberg–Lie algebra for the (nonlinear) second-order ODE:.

Second-Order ODEs with Three-Dimensional Vessiot–Guldberg–Lie Algebra

Examples

The well-known Milne–Pinney equation:

The Kummer–Schwarz equation:

SODE Lie Systems in the Plane

Conclusions