A partial Lagrangian method for dynamical systems

We develop a new approach termed as a discount free or partial Lagrangian method for construction of first integrals for dynamical systems of ordinary differential equations (ODEs). It is shown how one can utilize the Legendre transformation in a more general setting to provide the equivalence between a current value Hamiltonian and a partial or discount free Lagrangian when it exists. As a consequence, we develop a discount factor free Lagrangian framework to deduce reductions and closed-form solutions via first integrals for ODEs arising from economics by proving three important propositions. The approach is algorithmic and applies to many state variables of the Lagrangian. In order to show its effectiveness, we apply the method to models, one linear and two nonlinear, with one state variable. We obtain new exact solutions for the last model. The discount free Lagrangian naturally arises in economic growth theory and many other economic models when the control variables can be eliminated at the outset which is not always possible in optimal control theory applications of economics. We explain our method with the help of few widely used economic growth models. We point out the difference between this approach and the more general partial Hamiltonian method proposed earlier for a current value Hamiltonian (Naz et al. in Commun Nonlinear Sci Numer Simul 19:3600-3610, 2014) which is applicable in a general setting involving time, state, costate and control variables.