Higher-order Abel equations: Lagrangian formalism, first integrals and Darboux polynomials

TL;DR

This paper explores higher-order nonlinear Abel equations using geometric methods, establishing Lagrangian formulations, conserved quantities, and analyzing their structure via Darboux polynomials and Jacobi multipliers, including the general n-dimensional case.

Contribution

It provides the first Lagrangian formulations for higher-order Abel equations and links their integrability to Darboux polynomials and Jacobi multipliers.

Findings

Two non-natural Lagrangian formulations for second-order Abel equations

Explicit family of constants of motion derived

Extension of analysis to n-dimensional Abel equations

Abstract

A geometric approach is used to study a family of higher-order nonlinear Abel equations. The inverse problem of the Lagrangian dynamics is studied in the particular case of the second-order Abel equation and the existence of two alternative Lagrangian formulations is proved, both Lagrangians being of a non-natural class (neither potential nor kinetic term). These higher-order Abel equations are studied by means of their Darboux polynomials and Jacobi multipliers. In all the cases a family of constants of the motion is explicitly obtained. The general n-dimensional case is also studied.