2008/09/28 by M. C. Nucci, Nucci, M. C., A. M. Arthurs +1
Chemistry · Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Molecular spectroscopy and chirality #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.0809.4845
openalex publication_date 2008/09/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that given an ordinary differential equation of order four, it may be possible to determine a Lagrangian if the third derivative is absent (or eliminated) from the equation. This represents a subcase of Fels'conditions [M. E. Fels, The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations, Trans. Amer. Math. Soc. 348 (1996) 5007-5029] which ensure the existence and uniqueness of the Lagrangian in the case of a fourth-order equation. The key is the Jacobi last multiplier as in the case of a second-order equation. Two equations from a Number Theory paper by Hall, one of second and one of fourth order, will be used to exemplify the method. The known link between Jacobi last multiplier and Lie symmetries is also exploited. Finally the Lagrangian of two fourth-order equations drawn from Physics are determined with the same method.