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On sufficiency issues, first integrals and exact solutions of Uzawa-Lucas model with unskilled labor
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 |
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.
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. |
[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. |
[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. |
[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.
|
[5] |
B. U. Haq and I. Naeem,
First integrals and analytical solutions of some dynamical systems, Nonlinear Dynamics, 95 (2019), 1747-1765.
|
[6] |
V. V. Kozlov,
Integrability and nonintegrability in Hamiltonian mechanics, Russ. Math. Surveys, 38 (1983), 3-67,240.
|
[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.
|
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[19] |
R. Naz,
Characterization of approximate Partial Hamiltonian operators and related approximate first integrals, International Journal of Non-Linear Mechanics, 105 (2018), 158-164.
|
[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. |
[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. |
[22] |
S. Smale,
Topology and mechanics, Invent. Math., 10 (1970), 305-331.
doi: 10.1007/BF01418778. |
[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.![]() ![]() ![]() |
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. |
[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. |
[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. |
[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.
|
[5] |
B. U. Haq and I. Naeem,
First integrals and analytical solutions of some dynamical systems, Nonlinear Dynamics, 95 (2019), 1747-1765.
|
[6] |
V. V. Kozlov,
Integrability and nonintegrability in Hamiltonian mechanics, Russ. Math. Surveys, 38 (1983), 3-67,240.
|
[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.
|
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[19] |
R. Naz,
Characterization of approximate Partial Hamiltonian operators and related approximate first integrals, International Journal of Non-Linear Mechanics, 105 (2018), 158-164.
|
[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. |
[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. |
[22] |
S. Smale,
Topology and mechanics, Invent. Math., 10 (1970), 305-331.
doi: 10.1007/BF01418778. |
[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.![]() ![]() ![]() |
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