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A partial Lagrangian method for dynamical systems

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Abstract

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.

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Acknowledgments

FMM is thankful to Dr. Shahid Amjad Chaudhry, Rector, Lahore School of Economics, Pakistan, for his warm hospitality during which time this work was commenced. He also is grateful to the NRF of South Africa for enabling research support through grants. RN is thankful to Lahore School of Economics for providing funds to complete this work.

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Authors and Affiliations

  1. Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore, 53200, Pakistan

    R. Naz

  2. DST-NRF Centre of Excellence in Mathematical and Statistical Sciences, School of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg, Wits, 2050, South Africa

    F. M. Mahomed

  3. Department of Economics, Lahore School of Economics, Lahore, 53200, Pakistan

    A. Chaudhry

Authors
  1. R. Naz
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  2. F. M. Mahomed
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  3. A. Chaudhry
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Correspondence to R. Naz.

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Naz, R., Mahomed, F.M. & Chaudhry, A. A partial Lagrangian method for dynamical systems. Nonlinear Dyn 84, 1783–1794 (2016). https://doi.org/10.1007/s11071-016-2605-8

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