Higher-order Abel equations: Lagrangian formalism, Darboux polynomials and constants of the motion
José Cari\ nena, Partha Guha, and Manuel Ra\ nada
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Submission date: 23. Jul. 2009
published in: Nonlinearity, 22 (2009) 12, p. 2953-2969
DOI number (of the published article): 10.1088/0951-7715/22/12/008
with the following different title: Higher-order Abel equations : Lagrangian formalism, first integrals and Darboux polynomials
MSC-Numbers: 34A26, 34A34, 70H03
PACS-Numbers: 02.30.Hq, 45.20.Jj
Keywords and phrases: Higher-order Riccati equations, Darboux polynomials, Jacobi multipliers
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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 it is proved the existence of two alternative Lagrangian formulations, 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.