# American Institue of Mathematical Sciences

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

## 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

* 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.

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. [2] V. Dorodnitsyn, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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.
 [1] Rongmei Cao, Jiangong You. The existence of integrable invariant manifolds of Hamiltonian partial differential equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 227-234. doi: 10.3934/dcds.2006.16.227 [2] Azam Chaudhry, Rehana Naz. Closed-form solutions for the Lucas-Uzawa growth model with logarithmic utility preferences via the partial Hamiltonian approach. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 643-654. doi: 10.3934/dcdss.2018039 [3] Juan Dávila, Olivier Goubet. Partial regularity for a Liouville system. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2495-2503. doi: 10.3934/dcds.2014.34.2495 [4] Onur Alp İlhan. Solvability of some partial integral equations in Hilbert space. Communications on Pure & Applied Analysis, 2008, 7 (4) : 837-844. doi: 10.3934/cpaa.2008.7.837 [5] B. S. Goh, W. J. Leong, Z. Siri. Partial Newton methods for a system of equations. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 463-469. doi: 10.3934/naco.2013.3.463 [6] Răzvan M. Tudoran, Anania Gîrban. On the Hamiltonian dynamics and geometry of the Rabinovich system. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 789-823. doi: 10.3934/dcdsb.2011.15.789 [7] V. Barbu. Periodic solutions to unbounded Hamiltonian system. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 277-283. doi: 10.3934/dcds.1995.1.277 [8] Kim Knudsen, Mikko Salo. Determining nonsmooth first order terms from partial boundary measurements. Inverse Problems & Imaging, 2007, 1 (2) : 349-369. doi: 10.3934/ipi.2007.1.349 [9] Xiangjin Xu. Sub-harmonics of first order Hamiltonian systems and their asymptotic behaviors. Discrete & Continuous Dynamical Systems - B, 2003, 3 (4) : 643-654. doi: 10.3934/dcdsb.2003.3.643 [10] Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3807-3824. doi: 10.3934/dcds.2013.33.3807 [11] Cyril Joel Batkam. Homoclinic orbits of first-order superquadratic Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3353-3369. doi: 10.3934/dcds.2014.34.3353 [12] Fioralba Cakoni, Rainer Kress. Integral equations for inverse problems in corrosion detection from partial Cauchy data. Inverse Problems & Imaging, 2007, 1 (2) : 229-245. doi: 10.3934/ipi.2007.1.229 [13] Jian Zhang, Wen Zhang, Xiaoliang Xie. Existence and concentration of semiclassical solutions for Hamiltonian elliptic system. Communications on Pure & Applied Analysis, 2016, 15 (2) : 599-622. doi: 10.3934/cpaa.2016.15.599 [14] Nuno Costa Dias, Andrea Posilicano, João Nuno Prata. Self-adjoint, globally defined Hamiltonian operators for systems with boundaries. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1687-1706. doi: 10.3934/cpaa.2011.10.1687 [15] Emmanuel N. Barron, Rafal Goebel, Robert R. Jensen. The quasiconvex envelope through first-order partial differential equations which characterize quasiconvexity of nonsmooth functions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1693-1706. doi: 10.3934/dcdsb.2012.17.1693 [16] Masaki Hibino. Gevrey asymptotic theory for singular first order linear partial differential equations of nilpotent type — Part I — . Communications on Pure & Applied Analysis, 2003, 2 (2) : 211-231. doi: 10.3934/cpaa.2003.2.211 [17] Chungen Liu, Xiaofei Zhang. Subharmonic solutions and minimal periodic solutions of first-order Hamiltonian systems with anisotropic growth. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1559-1574. doi: 10.3934/dcds.2017064 [18] Michela Eleuteri, Pavel Krejčí. An asymptotic convergence result for a system of partial differential equations with hysteresis. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1131-1143. doi: 10.3934/cpaa.2007.6.1131 [19] Fanghua Lin, Chun Liu. Partial regularity of the dynamic system modeling the flow of liquid crystals. Discrete & Continuous Dynamical Systems - A, 1996, 2 (1) : 1-22. doi: 10.3934/dcds.1996.2.1 [20] Yves Achdou, Mathieu Laurière. On the system of partial differential equations arising in mean field type control. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 3879-3900. doi: 10.3934/dcds.2015.35.3879

2016 Impact Factor: 0.781

Article outline