A partial Hamiltonian approach for current value Hamiltonian systems |
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| Affiliation: | 1. Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore 53200, Pakistan;2. Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa;3. Department of Economics, Lahore School of Economics, Lahore 53200, Pakistan;1. A.I. Alikhanyan National Science Laboratory, 0036 Yerevan, Armenia;2. Departamento de Ciencias Exatas, Universidade Federal de Lavras, CP 3037, 37200-000 Lavras-MG, Brazil;3. Dipartimento di Scienza e Alta Tecnologia, Universitá degli Studi dell’Insubria, Via Valleggio 11, 22100 Como, Italy;4. I.N.F.N. Sezione di Milano, Via Celoria 16, 20133 Milano, Italy;5. Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR CNRS 6303, Université de Bourgogne, 21078 Dijon Cedex, France;6. Institute for Physical Research, 0203 Ashtarak-2, Armenia;1. Department of Mathematics, Texas A&M University-Texarkana, USA;2. Department of Mathematics, University of Central Florida, Orlando, USA;3. University of Palermo, Department of Mathematics, Palermo, Italy;1. Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China;2. Key Laboratory of Image Processing and Intelligent Control of Education Ministry of China, Wuhan 430074, China;3. Texas A & M University at Qatar, Doha 5825, Qatar;1. Saint Petersburg State University, St. Petersburg, Russia;2. University of L’Aquila, L’Aquila, Italy |
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| Abstract: | We develop a partial Hamiltonian framework to obtain reductions and closed-form solutions via first integrals of current value Hamiltonian systems of ordinary differential equations (ODEs). The approach is algorithmic and applies to many state and costate variables of the current value Hamiltonian. However, we apply the method to models with one control, one state and one costate variable to illustrate its effectiveness. The current value Hamiltonian systems arise in economic growth theory and other economic models. We explain our approach with the help of a simple illustrative example and then apply it to two widely used economic growth models: the Ramsey model with a constant relative risk aversion (CRRA) utility function and Cobb Douglas technology and a one-sector AK model of endogenous growth are considered. We show that our newly developed systematic approach can be used to deduce results given in the literature and also to find new solutions. |
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| Keywords: | Current value Hamiltonian Partial Hamiltonian approach Economic growth models |
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