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Conservation laws by symmetries and adjoint symmetries
Characterization of partial Hamiltonian operators and related first integrals
a. | Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore 53200, Pakistan |
b. | 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 |
We focus on partial Hamiltonian systems for the characterization of their operators and related first integrals. Firstly, it is shown that if an operator is a partial Hamiltonian operator which yields a first integral, then so does its evolutionary representative. Secondly, extra operator conditions are provided for a partial Hamiltonian operator in evolutionary form to yield a first integral. Thirdly, characterization of partial Hamiltonian operators and related first integral conditions are provided for the partial Hamiltonian system. Applications to mechanics are presented to illustrate the theory.
References:
[1] |
A. C. Chiang, Elements of Dynamic Optimization, McGraw Hill, New York, 1992. |
[2] |
V. Dorodnitsyn and R. Kozlov,
Invariance and first integrals of continuous and discrete Hamitonian equations, J. Eng. Math., 66 (2010), 253-270.
doi: 10.1007/s10665-009-9312-0. |
[3] |
A. H. Kara, F. M. Mahomed, I. Naeem and C. Wafo Soh,
Partial Noether operators and first integrals via partial Lagrangians, Math. Methods in the Applied Sciences, 30 (2007), 2079-2089.
doi: 10.1002/mma.939. |
[4] |
V. V. Kozlov,
Integrability and nonintegrability in Hamiltonian mechanics, Russ. Math. Surveys, 38 (1983), 1-76.
|
[5] |
P. G. L. Leach,
First integrals for the modified Emden equation $\ddot q+\alpha (t)\dot q+q^n=0$, J. Math. Phys, 26 (1985), 2510-2514.
doi: 10.1063/1.526766. |
[6] |
T. Levi-Civita,
Interpretazione gruppale degli integrali di un sistema canonico, Rend. Acc. Lincei, ser. Ⅲ, 8 (1899), 235-238.
|
[7] |
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.
|
[8] |
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. |
[9] |
J. E. 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. |
[10] |
R. Naz, F. M. Mahomed and A. Chaudhry,
A partial Hamiltonian approach for current value Hamiltonian systems, Commu. Nonlinear. Sci. Numer. Simulat., 19 (2014), 3600-3610.
doi: 10.1016/j.cnsns.2014.03.023. |
[11] |
R. Naz, A. Chaudhry and F. M. Mahomed,
Closed-form solutions for the Lucas-Uzawa model of economic growth via the partial Hamiltonian approach, Commu. Nonlinear. Sci. Numer. Simulat, 30 (2016), 299-306.
doi: 10.1016/j.cnsns.2015.06.033. |
[12] |
R. Naz,
The applications of the partial Hamiltonian approach to mechanics and other areas, International Journal of Non-Linear Mechanics, 86 (2016), 1-6.
|
[13] |
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. |
[14] |
P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-4350-2. |
[15] |
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. |
show all references
References:
[1] |
A. C. Chiang, Elements of Dynamic Optimization, McGraw Hill, New York, 1992. |
[2] |
V. Dorodnitsyn and R. Kozlov,
Invariance and first integrals of continuous and discrete Hamitonian equations, J. Eng. Math., 66 (2010), 253-270.
doi: 10.1007/s10665-009-9312-0. |
[3] |
A. H. Kara, F. M. Mahomed, I. Naeem and C. Wafo Soh,
Partial Noether operators and first integrals via partial Lagrangians, Math. Methods in the Applied Sciences, 30 (2007), 2079-2089.
doi: 10.1002/mma.939. |
[4] |
V. V. Kozlov,
Integrability and nonintegrability in Hamiltonian mechanics, Russ. Math. Surveys, 38 (1983), 1-76.
|
[5] |
P. G. L. Leach,
First integrals for the modified Emden equation $\ddot q+\alpha (t)\dot q+q^n=0$, J. Math. Phys, 26 (1985), 2510-2514.
doi: 10.1063/1.526766. |
[6] |
T. Levi-Civita,
Interpretazione gruppale degli integrali di un sistema canonico, Rend. Acc. Lincei, ser. Ⅲ, 8 (1899), 235-238.
|
[7] |
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.
|
[8] |
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. |
[9] |
J. E. 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. |
[10] |
R. Naz, F. M. Mahomed and A. Chaudhry,
A partial Hamiltonian approach for current value Hamiltonian systems, Commu. Nonlinear. Sci. Numer. Simulat., 19 (2014), 3600-3610.
doi: 10.1016/j.cnsns.2014.03.023. |
[11] |
R. Naz, A. Chaudhry and F. M. Mahomed,
Closed-form solutions for the Lucas-Uzawa model of economic growth via the partial Hamiltonian approach, Commu. Nonlinear. Sci. Numer. Simulat, 30 (2016), 299-306.
doi: 10.1016/j.cnsns.2015.06.033. |
[12] |
R. Naz,
The applications of the partial Hamiltonian approach to mechanics and other areas, International Journal of Non-Linear Mechanics, 86 (2016), 1-6.
|
[13] |
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. |
[14] |
P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-4350-2. |
[15] |
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. |
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