American Institute of Mathematical Sciences

Advanced
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

[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]

Rehana Naz, Fazal M Mahomed, Azam Chaudhry. First integrals of Hamiltonian systems: The inverse problem. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020121
[4]

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
[5]

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
[6]

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
[7]

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
[8]

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
[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]

Yuhki Hosoya. First-order partial differential equations and consumer theory. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1143-1167. doi: 10.3934/dcdss.2018065
[13]

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
[14]

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
[15]

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
[16]

Xing Huang, Wujun Lv. Stochastic functional Hamiltonian system with singular coefficients. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1257-1273. doi: 10.3934/cpaa.2020060
[17]

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
[18]

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
[19]

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
[20]

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

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (59)
  • HTML views (374)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences