Inverse Variational Problem for Nonlinear Dynamical Systems

In this paper we have chosen to work with two different approaches solving the inverse problem of calculus variation. The first approach is based on an integral representation Lagrangian function that uses equation motion while second one relies a generalization well known Noether's theorem and constructs directly from motion. As application provide some useful remarks for modified Emden-type then obtain results functions (i) cubic-quintic Duffing oscillator, (ii) Lienard-type oscillator (iii) Mathews-Lakshmanan oscillator. these oscillators were found be characterized by nonstandard Lagrangians except could also assign standard We used find indirect analytic (Lagrangian) three velocity-dependent equations (iv) Abraham-Lorentz (v) Lorentz (vi) Van der Pol For each dynamical systems (i)-(vi) calculated result Jacobi thereby provided method Hamiltonian without taking recourse use so-called Legendre transformation.