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

Fels’ conditions (Fels, M. E. 1996 Trans. Amer. Math. Soc. 348, 5007–5029. (doi:10.1090/ S0002-9947-96-01720-5)) 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.