On the inverse problem of calculus of variations for fourth-order equations

Abstract

)) ensure the existence and uniqueness of the Lagrangian in the case of a fourth-order equation. We show that when Fels' conditions are satisfied, the Lagrangian can be derived from the Jacobi last multiplier, as in the case of a secondorder equation. Indeed, we prove that if a Lagrangian exists for an equation of any even order, then it can be derived from the Jacobi last multiplier. Two equations from a Number Theory paper by Hall (Hall, R. R. 2002 J. Number Theory 93, 235-245. (doi:10.1006/jnth.2001.2719)), one of the second and one of the fourth order, will be used to exemplify the method. The known link between Jacobi last multiplier and Lie symmetries is also exploited. Finally, the Lagrangians of two fourth-order equations drawn from Physics are determined with the same method.Keywords: fourth-order ordinary differential equations; inverse problem of calculus of variations; Lagrangian; Jacobi last multiplier

On the inverse problem of the discrete calculus of variations

“…For an historical perspective on this subject we refer to [28][29][30] and references therein. Moreover, we observe that the Jacobi Last Multiplier can be used to find Lagrangians for differential equations of order higher than two [31] and for non-dissipative systems of second-order equations [32,33].…”

Conditionally integrable PT -symmetric hierarchies and near-Lax pairs of the KdV and KPII equations

“…A natural area of extension would be to symmetry analysis [32,33], which usually yields additional results and insights. Also, derivations of Lagrangian and Hamiltonian formulations for the traveling wave equations [34,35] is likely to be worthwhile to investigate other solutions and features of the  -symmetric NLPDEs considered here. Such Lagrangian formulations may be used to variationally construct both regular and embedded solutions [36,37].…”

An Improved Criterion for the Oscillation of Fourth-Order Differential Equations

“…Many papers have been concerned to the solution of the inverse problem of calculus of variations, namely finding a Lagrangian of differential equations. Also, the use of the Jacobi last multiplier and its connection with Lie theory, in order to find the Lagrangian for ordinary differential equations, can be found in [29].…”