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

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.

References in corpus (3)

• Jacobi's Last Multiplier and Lagrangians for Multidimensional Systems
• Symmetries of Toda Equations
• Symmetry, Equivalence and Integrable Classes of Abel Equations

Cited by in corpus (10)

• A direct approach to the construction of standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients
• Generating Finite Dimensional Integrable Nonlinear Dynamical Systems
• Nonlocal symmetries of Riccati and Abel chains and their similarity reductions
• Symmetries of nonlinear ordinary differential equations: the modified Emden equation as a case study
• A geometric approach to integrability of Abel differential equations
• Nonlocal symmetries of a class of scalar and coupled nonlinear ordinary differential equations of any order
• Generalized symmetries, first integrals, and exact solutions of chains of differential equations
• Order preserving contact transformations and dynamical symmetries of scalar and coupled Riccati and Abel chains
• Invariance of second order ordinary differential equations under two-dimensional affine subalgebras of EP Lie algebra
• Liouvillian and Analytic Integrability of a Generalized Gause System