Inverse Variational Problem for Nonstandard Lagrangians

In the mathematical physics literature the nonstandard Lagrangians (NSLs) were introduced in an ad hoc fashion rather than being derived from the solution of the inverse problem of variational calculus. We begin with the first integral of the equation of motion and solve the associated inverse problem to obtain some of the existing results for NSLs. In addition, we provide a number of alternative Lagrangian representations. The case studies envisaged by us include (i) the usual modified Emden-type equation, (ii) Emden-type equation with dissipative term quadratic in velocity, (iii) Lotka–Volterra model and (vi) a number of the generic equations for dissipative-like dynamical systems. Our method works for nonstandard Lagrangians corresponding to the usual action integral of mechanical systems but requires modification for those associated with the modified actions like S = ∫ a b e L ( x , x ˙ , t ) d t and S = ∫ a b L 1 - γ ( x , x ˙ , t ) d t because in the latter case one cannot construct expressions for the Jacobi integrals.