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On sufficiency issues, first integrals and exact solutions of Uzawa-Lucas model with unskilled labor
October  2020, 13(10): 2829-2840. doi: 10.3934/dcdss.2020121

## First integrals of Hamiltonian systems: The inverse problem

 1 Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore 53200, Pakistan 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 3 Department of Economics, Lahore School of Economics, Lahore 53200, Pakistan

* Corresponding author: Fazal M Mahomed

Received  January 2019 Revised  May 2019 Published  October 2019

There has, to date, been much focus on when a Hamiltonian operator or symmetry results in a first integral for Hamiltonian systems. Very little emphasis has been given to the inverse problem, viz. which operator arises from a first integral of a Hamiltonian system. In this note, we consider this problem with examples mainly taken from economic growth theory. We also provide an example from classical mechanics.

Citation: Rehana Naz, Fazal M Mahomed, Azam Chaudhry. First integrals of Hamiltonian systems: The inverse problem. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2829-2840. doi: 10.3934/dcdss.2020121
##### References:
 [1] V. I. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar [2] M. V. Berry and P. Shukla, Classical dynamics with curl forces, and motion driven by time-dependent flux, J. Phys. A: Math. Theor., 45 (2012), 305201, 18 pp. doi: 10.1088/1751-8113/45/30/305201.  Google Scholar [3] V. Dorodnitsyn and R. Kozlov, Invariance and first integrals of continuous and discrete Hamiltonian equations, Journal of Engineering Mathematics, 66 (2010), 253-270.  doi: 10.1007/s10665-009-9312-0.  Google Scholar [4] B. U. Haq and I. Naeem, First integrals and exact solutions of some compartmental disease models, Zeitschrift für Naturforschung A, 74 (2019), 293-304.   Google Scholar [5] B. U. Haq and I. Naeem, First integrals and analytical solutions of some dynamical systems, Nonlinear Dynamics, 95 (2019), 1747-1765.   Google Scholar [6] V. V. Kozlov, Integrability and nonintegrability in Hamiltonian mechanics, Russ. Math. Surveys, 38 (1983), 3-67,240.   Google Scholar [7] T. Levi-Civita, Interpretazione gruppale degli integrali di un sistema canonico, Rend. Acc. Lincei, s. 3$^a$, sem. 2$^o$ sem., 7 (1899), 235-238.   Google Scholar [8] F. M. Mahomed and J. A. G. Roberts, Characterization of Hamiltonian symmetries and their first integrals., International Journal of Non-Linear Mechanics, 74 (2015), 84-91.   Google Scholar [9] K. S. Mahomed and R. J. Moitsheki, First integrals of generalized Ermakov systems via the Hamiltonian formulation, International Journal of Modern Physics B, 30 (2016), 1640019, 12 pp. doi: 10.1142/S0217979216400191.  Google Scholar [10] J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Math. Phys., 5 (1974), 121-130.  doi: 10.1016/0034-4877(74)90021-4.  Google Scholar [11] R. Naz, F. M. Mahomed and A. Chaudhry, A partial Hamiltonian approach for current value Hamiltonian systems, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 3600-3610.  doi: 10.1016/j.cnsns.2014.03.023.  Google Scholar [12] R. Naz, F. M. Mahomed and A. Chaudhry, A partial Lagrangian method for dynamical systems, Nonlinear Dynamics, 84 (2016), 1783-1794.  doi: 10.1007/s11071-016-2605-8.  Google Scholar [13] R. Naz, A. Chaudhry and F. M. Mahomed, Closed-form solutions for the Lucas-Uzawa model of economic growth via the partial Hamiltonian approach, Communications in Nonlinear Science and Numerical Simulation, 30 (2016), 299-306.  doi: 10.1016/j.cnsns.2015.06.033.  Google Scholar [14] R. Naz and A. Chaudhry, Comparison of closed-form solutions for the Lucas-Uzawa model via the partial Hamiltonian approach and the classical approach, Mathematical Modelling and Analysis, 22 (2017), 464-483.  doi: 10.3846/13926292.2017.1323035.  Google Scholar [15] R. Naz and A. Chaudhry, Closed-form solutions of Lucas-Uzawa model with externalities via partial Hamiltonian approach, Computational and Applied Mathematics, 37 (2018), 5146-5161.  doi: 10.1007/s40314-018-0622-6.  Google Scholar [16] R. Naz and I. Naeem, The artificial Hamiltonian, first integrals, and closed-form solutions of dynamical systems for epidemics, Zeitschrift für Naturforschung A, 73 (2018), 323–330. Google Scholar [17] R. Naz, The applications of the partial Hamiltonian approach to mechanics and other areas, International Journal of Non-linear Mechanics, 86 (2016), 1-6.   Google Scholar [18] R. Naz and F. M. Mahomed, Characterization of partial Hamiltonian operators and related first integrals, Discrete & Continuous Dynamical Systems-Series S (DCDS-S), 11 (2018), 723-734.  doi: 10.3934/dcdss.2018045.  Google Scholar [19] R. Naz, Characterization of approximate Partial Hamiltonian operators and related approximate first integrals, International Journal of Non-Linear Mechanics, 105 (2018), 158-164.   Google Scholar [20] P. J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, 107. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4350-2.  Google Scholar [21] G. Saccomandi and R. Vitolo, A translation of the T. Levi-Civita paper: Interpretazione Gruppale degli integrali di un Sistema Canonico, Regul. Chaotic Dyn., 17 (2012), 105–112, arXiv: 1201.2388v1. doi: 10.1134/S1560354712010091.  Google Scholar [22] S. Smale, Topology and mechanics, Invent. Math., 10 (1970), 305-331.  doi: 10.1007/BF01418778.  Google Scholar [23] E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: With an Introduction to the Problem of Three Bodies, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988.  doi: 10.1017/CBO9780511608797.  Google Scholar

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##### References:
 [1] V. I. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar [2] M. V. Berry and P. Shukla, Classical dynamics with curl forces, and motion driven by time-dependent flux, J. Phys. A: Math. Theor., 45 (2012), 305201, 18 pp. doi: 10.1088/1751-8113/45/30/305201.  Google Scholar [3] V. Dorodnitsyn and R. Kozlov, Invariance and first integrals of continuous and discrete Hamiltonian equations, Journal of Engineering Mathematics, 66 (2010), 253-270.  doi: 10.1007/s10665-009-9312-0.  Google Scholar [4] B. U. Haq and I. Naeem, First integrals and exact solutions of some compartmental disease models, Zeitschrift für Naturforschung A, 74 (2019), 293-304.   Google Scholar [5] B. U. Haq and I. Naeem, First integrals and analytical solutions of some dynamical systems, Nonlinear Dynamics, 95 (2019), 1747-1765.   Google Scholar [6] V. V. Kozlov, Integrability and nonintegrability in Hamiltonian mechanics, Russ. Math. Surveys, 38 (1983), 3-67,240.   Google Scholar [7] T. Levi-Civita, Interpretazione gruppale degli integrali di un sistema canonico, Rend. Acc. Lincei, s. 3$^a$, sem. 2$^o$ sem., 7 (1899), 235-238.   Google Scholar [8] F. M. Mahomed and J. A. G. Roberts, Characterization of Hamiltonian symmetries and their first integrals., International Journal of Non-Linear Mechanics, 74 (2015), 84-91.   Google Scholar [9] K. S. Mahomed and R. J. Moitsheki, First integrals of generalized Ermakov systems via the Hamiltonian formulation, International Journal of Modern Physics B, 30 (2016), 1640019, 12 pp. doi: 10.1142/S0217979216400191.  Google Scholar [10] J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Math. Phys., 5 (1974), 121-130.  doi: 10.1016/0034-4877(74)90021-4.  Google Scholar [11] R. Naz, F. M. Mahomed and A. Chaudhry, A partial Hamiltonian approach for current value Hamiltonian systems, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 3600-3610.  doi: 10.1016/j.cnsns.2014.03.023.  Google Scholar [12] R. Naz, F. M. Mahomed and A. Chaudhry, A partial Lagrangian method for dynamical systems, Nonlinear Dynamics, 84 (2016), 1783-1794.  doi: 10.1007/s11071-016-2605-8.  Google Scholar [13] R. Naz, A. Chaudhry and F. M. Mahomed, Closed-form solutions for the Lucas-Uzawa model of economic growth via the partial Hamiltonian approach, Communications in Nonlinear Science and Numerical Simulation, 30 (2016), 299-306.  doi: 10.1016/j.cnsns.2015.06.033.  Google Scholar [14] R. Naz and A. Chaudhry, Comparison of closed-form solutions for the Lucas-Uzawa model via the partial Hamiltonian approach and the classical approach, Mathematical Modelling and Analysis, 22 (2017), 464-483.  doi: 10.3846/13926292.2017.1323035.  Google Scholar [15] R. Naz and A. Chaudhry, Closed-form solutions of Lucas-Uzawa model with externalities via partial Hamiltonian approach, Computational and Applied Mathematics, 37 (2018), 5146-5161.  doi: 10.1007/s40314-018-0622-6.  Google Scholar [16] R. Naz and I. Naeem, The artificial Hamiltonian, first integrals, and closed-form solutions of dynamical systems for epidemics, Zeitschrift für Naturforschung A, 73 (2018), 323–330. Google Scholar [17] R. Naz, The applications of the partial Hamiltonian approach to mechanics and other areas, International Journal of Non-linear Mechanics, 86 (2016), 1-6.   Google Scholar [18] R. Naz and F. M. Mahomed, Characterization of partial Hamiltonian operators and related first integrals, Discrete & Continuous Dynamical Systems-Series S (DCDS-S), 11 (2018), 723-734.  doi: 10.3934/dcdss.2018045.  Google Scholar [19] R. Naz, Characterization of approximate Partial Hamiltonian operators and related approximate first integrals, International Journal of Non-Linear Mechanics, 105 (2018), 158-164.   Google Scholar [20] P. J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, 107. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4350-2.  Google Scholar [21] G. Saccomandi and R. Vitolo, A translation of the T. Levi-Civita paper: Interpretazione Gruppale degli integrali di un Sistema Canonico, Regul. Chaotic Dyn., 17 (2012), 105–112, arXiv: 1201.2388v1. doi: 10.1134/S1560354712010091.  Google Scholar [22] S. Smale, Topology and mechanics, Invent. Math., 10 (1970), 305-331.  doi: 10.1007/BF01418778.  Google Scholar [23] E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: With an Introduction to the Problem of Three Bodies, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988.  doi: 10.1017/CBO9780511608797.  Google Scholar
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