# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems - S, 2018, 11 (4) : 735-746. doi: 10.3934/dcdss.2018046
##### References:
 [1] L. M. Berkovich, Factorisations and Transformations of Differential Equations, Methods and Applications. Regular and Chaotic Dynamics, Oxford, 2002. [2] C. Bertoni, F. 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. [3] G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer-Verlag, 1989. doi: 10.1007/978-1-4757-4307-4. [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. [5] L. G. S. Duarte, S. E. S. Duarte, L. 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. [6] V. P. Ermakov, Univ. Izv. Kiev, 20 (1880). [7] F. Finelli, G. P. Vacca and G. Venturi, Phys. Rev. D, 58 (1998), 103514. [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. [9] N. H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations, vols, Ⅰ-Ⅲ, 1994. [10] N. H. Ibragimov, A. H. Kara and F. M. Mahomed, Lie-Backlund and Noether symmetries with applications, Nonlinear Dynamics, 15 (1998), 115-136.  doi: 10.1023/A:1008240112483. [11] A. H. Kara, F. M. Mahomed, I. 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. [12] J. M. Levy-leblond, Conservation laws for Gauge Invariant Lagrangians in Classical Mechanic, American Journal of Physics, 11 (1978), 249-258. [13] P. G. Leach, Generalized Ermakov systems, Physics Letters, 158 (1991), 102-106.  doi: 10.1016/0375-9601(91)90908-Q. [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. [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. [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. [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. [18] P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, 1986. doi: 10.1007/978-1-4684-0274-2. [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. [20] L. V. Ovsiannikov, Group Analysis of Differential Equations, Moscow: Nauka, 1978. [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. [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. [23] C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases, Cambridge University Press, Cambridge, 2001. [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. [25] L. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Oxford University Press, Oxford, 2003. [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. [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.

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##### References:
 [1] L. M. Berkovich, Factorisations and Transformations of Differential Equations, Methods and Applications. Regular and Chaotic Dynamics, Oxford, 2002. [2] C. Bertoni, F. 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. [3] G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer-Verlag, 1989. doi: 10.1007/978-1-4757-4307-4. [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. [5] L. G. S. Duarte, S. E. S. Duarte, L. 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. [6] V. P. Ermakov, Univ. Izv. Kiev, 20 (1880). [7] F. Finelli, G. P. Vacca and G. Venturi, Phys. Rev. D, 58 (1998), 103514. [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. [9] N. H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations, vols, Ⅰ-Ⅲ, 1994. [10] N. H. Ibragimov, A. H. Kara and F. M. Mahomed, Lie-Backlund and Noether symmetries with applications, Nonlinear Dynamics, 15 (1998), 115-136.  doi: 10.1023/A:1008240112483. [11] A. H. Kara, F. M. Mahomed, I. 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. [12] J. M. Levy-leblond, Conservation laws for Gauge Invariant Lagrangians in Classical Mechanic, American Journal of Physics, 11 (1978), 249-258. [13] P. G. Leach, Generalized Ermakov systems, Physics Letters, 158 (1991), 102-106.  doi: 10.1016/0375-9601(91)90908-Q. [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. [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. [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. [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. [18] P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, 1986. doi: 10.1007/978-1-4684-0274-2. [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. [20] L. V. Ovsiannikov, Group Analysis of Differential Equations, Moscow: Nauka, 1978. [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. [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. [23] C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases, Cambridge University Press, Cambridge, 2001. [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. [25] L. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Oxford University Press, Oxford, 2003. [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. [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.
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