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August  2018, 11(4): 723-734. doi: 10.3934/dcdss.2018045

Characterization of partial Hamiltonian operators and related first integrals

Rehana Naz a, and  Fazal M. Mahomed b,,

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

* Corresponding author: Fazal M. Mahomed

Received  November 2016 Revised  April 2017 Published  November 2017

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.

Keywords: Partial Hamiltonian system, partial Hamiltonian operators, first integral.
Mathematics Subject Classification: 37K05, 70S05, 70S10.
Citation: Rehana Naz, Fazal M. Mahomed. Characterization of partial Hamiltonian operators and related first integrals. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 723-734. doi: 10.3934/dcdss.2018045
References:
[1]

A. C. Chiang, Elements of Dynamic Optimization, McGraw Hill, New York, 1992. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

V. V. Kozlov, Integrability and nonintegrability in Hamiltonian mechanics, Russ. Math. Surveys, 38 (1983), 1-76.   Google Scholar

[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.  Google Scholar

[6]

T. Levi-Civita, Interpretazione gruppale degli integrali di un sistema canonico, Rend. Acc. Lincei, ser. Ⅲ, 8 (1899), 235-238.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[14]

P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4350-2.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

A. C. Chiang, Elements of Dynamic Optimization, McGraw Hill, New York, 1992. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

V. V. Kozlov, Integrability and nonintegrability in Hamiltonian mechanics, Russ. Math. Surveys, 38 (1983), 1-76.   Google Scholar

[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.  Google Scholar

[6]

T. Levi-Civita, Interpretazione gruppale degli integrali di un sistema canonico, Rend. Acc. Lincei, ser. Ⅲ, 8 (1899), 235-238.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[14]

P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4350-2.  Google Scholar

[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.  Google Scholar

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