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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. NazF. 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. NazF. 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. NazA. 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

show all references

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. NazF. 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. NazF. 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. NazA. 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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