An Old Method of Jacobi to Find Lagrangians

In a recent paper by Ibragimov [N. H. Ibragimov, Invariant Lagrangians and a new method of integration of nonlinear equations, J. Math. Anal. Appl. 304 (2005) 212--235] a method was presented in order to find Lagrangians of certain second-order ordinary differential equations admitting a two-dimensional Lie symmetry algebra. We present a method devised by Jacobi which enables to derive (many) Lagrangians of any second-order differential equation. The method is based on the search of the Jacobi Last Multipliers of the equations. We exemplify the simplicity and elegance of Jacobi's method by applying it to the same two equations as did Ibragimov. We show that the Lagrangians obtained by Ibragimov are particular cases of some of the many Lagrangians that can be obtained by Jacobi's method.