Lie symmetries, Lagrangians and Hamiltonian framework of a class of nonlinear nonautonomous equations

The method of Lie symmetries and the Jacobi Last Multiplier is used to study certain aspects of nonautonomous ordinary differential equations. Specifically we derive Lagrangians for a number of cases such as the Langmuir-Blodgett equation, the Langmuir-Bogulavski equation, the Lane-Emden-Fowler equation and the Thomas-Fermi equation by using the Jacobi Last Multiplier. By combining a knowledge of the last multiplier together with the Lie symmetries of the corresponding equations we explicitly construct first integrals for the Langmuir-Bogulavski equation q¨+53tq̇-t-5/3q-1/2=0 and the Lane-Emden-Fowler equation. These first integrals together with their corresponding Hamiltonains are then used to study time-dependent integrable systems. The use of the Poincaré-Cartan form allows us to find the conjugate Noetherian invariants associated with the invariant manifold.