August  2018, 11(4): 735-746. doi: 10.3934/dcdss.2018046

New conservation forms and Lie algebras of Ermakov-Pinney equation

1. 

Istanbul Technical University, Faculty of Science and Letters, Department of Mathematical Engineering, 34469 Maslak, Istanbul, Turkey

2. 

Istanbul Technical University, Faculty of Civil Engineering, Division of Mechanics, 34469 Maslak, Istanbul, Turkey

* Corresponding author: Teoman Özer

Received  December 2016 Revised  May 2017 Published  November 2017

In this study, we investigate first integrals and exact solutions of the Ermakov-Pinney equation. Firstly, the Lagrangian for the equation is constructed and then the determining equations are obtained based on the Lagrangian approach. Noether symmetry classification is implemented, the first integrals, conservation laws are obtained and classified. This classification includes Noether symmetries and first integrals with respect to different choices of external potential function. Furthermore, the time independent integrals and analytical solutions are obtained by using the modified Prelle-Singer procedure as a different approach. Additionally, for the investigation of conservation laws of the equation, we present the mathematical connections between the λ-symmetries, Lie point symmetries and the modified Prelle-Singer procedure. Finally, new Lagrangian and Hamiltonian forms of the equation are determined.

Citation: Özlem Orhan, Teoman Özer. New conservation forms and Lie algebras of Ermakov-Pinney equation. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 735-746. doi: 10.3934/dcdss.2018046
References:
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L. M. Berkovich, Factorisations and Transformations of Differential Equations, Methods and Applications. Regular and Chaotic Dynamics, Oxford, 2002. Google Scholar

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C. BertoniF. Finelli and G. Venturi, Adiabatic invariants and scalar fields in a de Sitter space-time, Phys. Lett. A, 237 (1998), 331-336.  doi: 10.1016/S0375-9601(97)00707-X.  Google Scholar

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L. G. S. DuarteS. E. S. DuarteL. A. C. P. da Mota and J. E. F. Skea, Solving second-order ordinary differential equations by extending the Prelle-Singer method, J. Phys. A:Math. Gen., 34 (2001), 3015-3024.  doi: 10.1088/0305-4470/34/14/308.  Google Scholar

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V. P. Ermakov, Univ. Izv. Kiev, 20 (1880). Google Scholar

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F. Finelli, G. P. Vacca and G. Venturi, Phys. Rev. D, 58 (1998), 103514. Google Scholar

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N. H. IbragimovA. H. Kara and F. M. Mahomed, Lie-Backlund and Noether symmetries with applications, Nonlinear Dynamics, 15 (1998), 115-136.  doi: 10.1023/A:1008240112483.  Google Scholar

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P. G. Leach, Generalized Ermakov systems, Physics Letters, 158 (1991), 102-106.  doi: 10.1016/0375-9601(91)90908-Q.  Google Scholar

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M. Molati and C. M. Khalique, Lie symmetry analysis of the time-variable coefficient B-BBM equation, Advances in Difference Equations, 2012 (2012), 8pp. doi: 10.1186/1687-1847-2012-212.  Google Scholar

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E. Noether, Invariante Variationsprobleme, Nachr. König. Gesell. Wissen., Göttingen, Math. -Phys. Kl. Heft, 2 (1918), 235–257. English translation in Transport Theory and Statistical Physics, 1 (1971), 186–207 doi: 10.1080/00411457108231446.  Google Scholar

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M. C. Nucci and P. G. L. Leach, Jacobi's last multiplier and the complete symmetry group of the Ermakov-Pinney equation, Journal of Nonlinear Mathematical Physics, Oxford, 12 (2005), 305-320.  doi: 10.2991/jnmp.2005.12.2.10.  Google Scholar

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P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, 1986. doi: 10.1007/978-1-4684-0274-2.  Google Scholar

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Ö. Orhan and T. Özer, Linearization properties, first integrals, nonlocal transformation for heat transfer equation, Int. J. Mod. Phys. B, 30 (2016), 1640024, 12 pp. doi: 10.1142/S0217979216400245.  Google Scholar

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L. V. Ovsiannikov, Group Analysis of Differential Equations, Moscow: Nauka, 1978.  Google Scholar

[21]

T. Özer, On symmetry group properties and general similarity forms of the Benney equations in the Lagrangian variables, Journal of Computational and Applied Mathematics, 169 (2004), 297-313.  doi: 10.1016/j.cam.2003.12.027.  Google Scholar

[22]

T. Özer, Symmetry group analysis and similarity solutions of variant nonlinear long wave equations, Chaos, Solitons Fractals, 33 (2008), 722-730.  doi: 10.1016/j.chaos.2007.01.023.  Google Scholar

[23]

C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases, Cambridge University Press, Cambridge, 2001. Google Scholar

[24]

E. Pinney, The nonlinear differential equation $\ddot{y}(x)+p(x)y+cy^{-3}=0$, Proceedings of the American Mathematical Society, Oxford, 1 (1950), p681. doi: 10.2307/2032300.  Google Scholar

[25]

L. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Oxford University Press, Oxford, 2003.  Google Scholar

[26]

M. Prelle and M. Singer, Elemantary first integrals of differential equations, Trans. Am. Math. Soc., 279 (1983), 215-219.  doi: 10.1090/S0002-9947-1983-0704611-X.  Google Scholar

[27]

J. R. Ray and J. L. Reid, Noether's theorem and Ermakov systems for nonlinear equations of motion, Physics Letters, 59 (1980), 134-140.  doi: 10.1007/BF02902329.  Google Scholar

show all references

References:
[1]

L. M. Berkovich, Factorisations and Transformations of Differential Equations, Methods and Applications. Regular and Chaotic Dynamics, Oxford, 2002. Google Scholar

[2]

C. BertoniF. Finelli and G. Venturi, Adiabatic invariants and scalar fields in a de Sitter space-time, Phys. Lett. A, 237 (1998), 331-336.  doi: 10.1016/S0375-9601(97)00707-X.  Google Scholar

[3]

G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer-Verlag, 1989. doi: 10.1007/978-1-4757-4307-4.  Google Scholar

[4]

V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan, On the Lagrangian and Hamiltonian description of the damped linear harmonic oscillator, J. Math. Phys., 48 (2007), 032701, 12 pp. doi: 10.1063/1.2711375.  Google Scholar

[5]

L. G. S. DuarteS. E. S. DuarteL. A. C. P. da Mota and J. E. F. Skea, Solving second-order ordinary differential equations by extending the Prelle-Singer method, J. Phys. A:Math. Gen., 34 (2001), 3015-3024.  doi: 10.1088/0305-4470/34/14/308.  Google Scholar

[6]

V. P. Ermakov, Univ. Izv. Kiev, 20 (1880). Google Scholar

[7]

F. Finelli, G. P. Vacca and G. Venturi, Phys. Rev. D, 58 (1998), 103514. Google Scholar

[8]

R. T. Herbst, The equivalence of linear and nonlinear differential equations, Proc. Am. Math. Soc., 7 (1956), 95-97.  doi: 10.1090/S0002-9939-1956-0076115-0.  Google Scholar

[9]

N. H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations, vols, Ⅰ-Ⅲ, 1994. Google Scholar

[10]

N. H. IbragimovA. H. Kara and F. M. Mahomed, Lie-Backlund and Noether symmetries with applications, Nonlinear Dynamics, 15 (1998), 115-136.  doi: 10.1023/A:1008240112483.  Google Scholar

[11]

A. H. KaraF. M. MahomedI. Naeem and C. Wafo Soh, Partial Noether operators and first integrals via partial Lagrangians, Math. Meth. Appl. Sci., Oxford, 30 (2007), 2079-2089.  doi: 10.1002/mma.939.  Google Scholar

[12]

J. M. Levy-leblond, Conservation laws for Gauge Invariant Lagrangians in Classical Mechanic, American Journal of Physics, 11 (1978), 249-258.   Google Scholar

[13]

P. G. Leach, Generalized Ermakov systems, Physics Letters, 158 (1991), 102-106.  doi: 10.1016/0375-9601(91)90908-Q.  Google Scholar

[14]

M. Molati and C. M. Khalique, Lie symmetry analysis of the time-variable coefficient B-BBM equation, Advances in Difference Equations, 2012 (2012), 8pp. doi: 10.1186/1687-1847-2012-212.  Google Scholar

[15]

C. Muriel and J. L. Romero, First integrals, integrating factors and $λ$-symmetries of second-order differential equations, Journal of Physics A, 42 (2009), 365207, 17 pp. doi: 10.1088/1751-8113/42/36/365207.  Google Scholar

[16]

E. Noether, Invariante Variationsprobleme, Nachr. König. Gesell. Wissen., Göttingen, Math. -Phys. Kl. Heft, 2 (1918), 235–257. English translation in Transport Theory and Statistical Physics, 1 (1971), 186–207 doi: 10.1080/00411457108231446.  Google Scholar

[17]

M. C. Nucci and P. G. L. Leach, Jacobi's last multiplier and the complete symmetry group of the Ermakov-Pinney equation, Journal of Nonlinear Mathematical Physics, Oxford, 12 (2005), 305-320.  doi: 10.2991/jnmp.2005.12.2.10.  Google Scholar

[18]

P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, 1986. doi: 10.1007/978-1-4684-0274-2.  Google Scholar

[19]

Ö. Orhan and T. Özer, Linearization properties, first integrals, nonlocal transformation for heat transfer equation, Int. J. Mod. Phys. B, 30 (2016), 1640024, 12 pp. doi: 10.1142/S0217979216400245.  Google Scholar

[20]

L. V. Ovsiannikov, Group Analysis of Differential Equations, Moscow: Nauka, 1978.  Google Scholar

[21]

T. Özer, On symmetry group properties and general similarity forms of the Benney equations in the Lagrangian variables, Journal of Computational and Applied Mathematics, 169 (2004), 297-313.  doi: 10.1016/j.cam.2003.12.027.  Google Scholar

[22]

T. Özer, Symmetry group analysis and similarity solutions of variant nonlinear long wave equations, Chaos, Solitons Fractals, 33 (2008), 722-730.  doi: 10.1016/j.chaos.2007.01.023.  Google Scholar

[23]

C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases, Cambridge University Press, Cambridge, 2001. Google Scholar

[24]

E. Pinney, The nonlinear differential equation $\ddot{y}(x)+p(x)y+cy^{-3}=0$, Proceedings of the American Mathematical Society, Oxford, 1 (1950), p681. doi: 10.2307/2032300.  Google Scholar

[25]

L. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Oxford University Press, Oxford, 2003.  Google Scholar

[26]

M. Prelle and M. Singer, Elemantary first integrals of differential equations, Trans. Am. Math. Soc., 279 (1983), 215-219.  doi: 10.1090/S0002-9947-1983-0704611-X.  Google Scholar

[27]

J. R. Ray and J. L. Reid, Noether's theorem and Ermakov systems for nonlinear equations of motion, Physics Letters, 59 (1980), 134-140.  doi: 10.1007/BF02902329.  Google Scholar

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