Using an old method of Jacobi to derive Lagrangians: a nonlinear dynamical system with variable coefficients

We present a method devised by Jacobi to derive Lagrangians of any second-order differential equation: it consists in finding a Jacobi Last Multiplier. We apply Jacobi's method to an equation that encompasses the motion of a particle subject to a force quadratic in the velocity. In particular, we illustrate the easiness and the power of Jacobi's method by applying it to the same equation studied by Musielak et al. with their own method (Z. E. Musielak, D. Roy and L. D. Swift, Chaos, Solitons & Fractals, 38 (2008) 894). While they were able to find one particular Lagrangian after lengthy calculations, Jacobi Last Multiplier method yields two different Lagrangians (and many others), of which one is that found by Musielak et al., and the other(s) is(are) quite new.