# American Institute of Mathematical Sciences

August  2017, 22(6): 2465-2478. doi: 10.3934/dcdsb.2017126

## Chiellini integrability condition, planar isochronous systems and Hamiltonian structures of Liénard equation

 1 Department of Physics, Surendranath College, 24/2 Mahatma Gandhi Road, Kolkata -700009, India 2 S.N. Bose National Centre for Basic Sciences, JD Block, Sector Ⅲ, Salt Lake, Kolkata -700098, India

A. Ghose Choudhury, E-mail address: aghosechoudhury@gmail.com

Received  June 2016 Revised  February 2017 Published  March 2017

Using a novel transformation involving the Jacobi Last Multiplier (JLM) we derive an old integrability criterion due to Chiellini for the Liénard equation. By combining the Chiellini condition for integrability and Jacobi's Last Multiplier the Lagrangian and Hamiltonian of the Liénard equation is derived. We also show that the Kukles equation is the only equation in the Liénard family which satisfies both the Chiellini integrability and the Sabatini criterion for isochronicity conditions. In addition we examine this result by mapping the Liénard equation to a harmonic oscillator equation using tacitly Chiellini's condition. Finally we provide a metriplectic and complex Hamiltonian formulation of the Liénard equation through the use of Chiellini condition for integrability.

Citation: A. Ghose Choudhury, Partha Guha. Chiellini integrability condition, planar isochronous systems and Hamiltonian structures of Liénard equation. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2465-2478. doi: 10.3934/dcdsb.2017126
##### References:
 [1] I. Bandić, Sur le critére intégrabilité de léquation différentielle généralis de Liénard, Bollettino dell'Unione Matematica Italiana, 16 (1961), 59-67. [2] A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden and T. Ratiu, The Euler-Poincaré equations and double Bracket dissipation, Comm. Math. Phys., 175 (1996), 1-42. [3] A. M. Bloch, P. J. Morrison and T. S. Ratiu, Gradient flows in the normal and Kähler metrics and triple bracket generated metriplectic systems, Recent Trends in Dynamical Systems, Springer, Proceedings in Mathematics and Statistics, 35 (2013), 371-415. [4] A. Chiellini, Sullíntegrazione della equazione differenziale y' + Py2 + Qy3 = 0, Bollettino della Unione Matem-atica Italiana, 10 (1931), 301-307. [5] C. Christopher and J. Devlin, On the classification of Liénard systems with amplitudeindependent periods, J. Differential Equations, 200 (2004), 1-17. [6] E. Kamke, Differentialgleichungen: Lösungsmethoden und Lösungen, Neunte Auflage. Mit einem Vorwort von Detlef Kamke. B. G. Teubner, Stuttgart, 1977. [7] A. Ghose Choudhury and P. Guha, On isochronous cases of the Cherkas system and Jacobi's last multiplier, J. Phys. A: Math. Theor. 43 (2010), 125202, 12pp. [8] A. Ghose Choudhury and P. Guha, An analytic technique for the solutions of nonlinear oscillators with damping using the Abel Equation, to appear in Discontinuity, Nonlinearity and Complexity. [9] A. Ghose Choudhury, P. Guha and B. Khanra, On the Jacobi last multiplier, integrating factors and the Lagrangian formulation of differential equations of the Painlevé-Gambier classification, J. Math. Anal. Appl., 360 (2009), 651-664. [10] M. Grmela, Hamiltonian extended thermodynamics, J. Phys. A: Math. Gen., 23 (1990), 3341-3351. [11] P. Guha and A. Ghose Choudhury, The Jacobi last multiplier and isochronicity of Liénard type systems, Rev. Math. Phys., 25 (2013), 1330009, 31pp. [12] P. Guha, Metriplectic structure, Leibniz dynamics and dissipative systems, J. Math. Anal. Appl., 326 (2007), 121-136. [13] T. Harko, F. S. N. Lobo and M. K. Mak, A class of exact solutions of the Liénard type ordinary non-linear differential equation arXiv: 1302.0836v3[math-ph]. [14] T. Harko, F. S. N. Lobo and M. K. Mak, A Chiellini type integrability condition for the generalized first kind Abel differential equation, Universal Journal of Applied Mathematics, 1 (2013), 101-104. [15] C. Jacobi, Sul principio dellúltimo moltiplicatore, e suo uso come nuovo principio generale di meccanica, Giornale Arcadico di Scienze, Lettere ed Arti 99 (1844), 129-146. [16] C. Jacobi, A. Clebsch and C. Brockhardt, Jacobi's Lectures on Dynamics, Texts and Readings in Mathematics, Hindustan Book Agency, 2009. [17] A. N. Kaufman, Dissipative Hamiltonian systems: A unifying principle. Phys. Lett.A, 100 (1984), 419-422. [18] A. Lichnerowicz, Les variétés de Poisson et leurs algébres de Lie associées, J. Differential Geom., 12 (1977), 253-300. [19] S. C. Mancas and H. C. Rosu, Integrable dissipative nonlinear second order differential equations via factorizations and Abel equations, Phys. Lett. A, 377 (2013), 1234-1238. [20] S. C. Mancas and H. C. Rosu, Integrable equations with Ermakov-Pinney nonlinearities and Chiellini damping, Appl. Math. Comp., 259 (2015), 1-11. [21] M. K. Mak, H. W. Chan and T. Harko, Solutions generating technique for Abel-type nonlinear ordinary differential equations, Comput. Math. Appl., 41 (2001), 1395-1401. [22] P. J. Morrison, A paradigm for joined Hamiltonian and dissipative systems, Physica D, 18 (1986), 410-419. [23] P. J. Morrison, Thoughts on brackets and dissipation: Old and new, J. Phys.: Conf. Ser., 169 (2009), 012006. [24] M. C. Nucci and P. G. L. Leach, Jacobi's last multiplier and symmetries for the Kepler problem plus a lineal story, J. Phys. A: Math. Gen., 37 (2004), 7743-7753. [25] M. C. Nucci and P. G. L. Leach, The Jacobi's Last Multiplier and its applications in mechanics, Phys. Scr., 78 (2008), 065011. [26] M. C. Nucci and K. M. Tamizhmani, Lagrangians for dissipative nonlinear oscillators: The method of Jacobi last multiplier, Journal of Nonlinear Mathematical Physics, 17 (2010), 167-178. [27] S. G. Rajeev, A canonical formulation of dissipative mechanics using complex-valued Hamiltonians, Ann. Physics, 322 (2007), 1541-1555. [28] B. S. Madhava Rao, On the reduction of dynamical equations to the Lagrangian form, Proc. Benaras Math. Soc. (N.S.), 2 (1940), 53-59. [29] A. Raouf Chouikha, Isochronous centers of Lienard type equations and applications, J. Math. Anal. Appl., 331 (2007), 358-376. [30] H. C. Rosu, S. C. Mancas and P. Chen, Barotropic FRW cosmolog ies with Chiellini damping, Phys. Lett. A, 379 (2015), 882-887. [31] M. Sabatini, On the period Function of Liénard Systems, J. Diff. Eqns., 152 (1999), 467-487. [32] T. Shah, R. Chattopadhyay, K. Vaidya and Sagar Chakraborty, Conservative perturbation theory for nonconservative systems, Phys. Rev. E, 92 (2015), 062927. [33] I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Birkhäuser-Verlag, Basel, 1994. [34] E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge Mathematical Library. Cambridge University Press,, Cambridge, 1988.

show all references

##### References:
 [1] I. Bandić, Sur le critére intégrabilité de léquation différentielle généralis de Liénard, Bollettino dell'Unione Matematica Italiana, 16 (1961), 59-67. [2] A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden and T. Ratiu, The Euler-Poincaré equations and double Bracket dissipation, Comm. Math. Phys., 175 (1996), 1-42. [3] A. M. Bloch, P. J. Morrison and T. S. Ratiu, Gradient flows in the normal and Kähler metrics and triple bracket generated metriplectic systems, Recent Trends in Dynamical Systems, Springer, Proceedings in Mathematics and Statistics, 35 (2013), 371-415. [4] A. Chiellini, Sullíntegrazione della equazione differenziale y' + Py2 + Qy3 = 0, Bollettino della Unione Matem-atica Italiana, 10 (1931), 301-307. [5] C. Christopher and J. Devlin, On the classification of Liénard systems with amplitudeindependent periods, J. Differential Equations, 200 (2004), 1-17. [6] E. Kamke, Differentialgleichungen: Lösungsmethoden und Lösungen, Neunte Auflage. Mit einem Vorwort von Detlef Kamke. B. G. Teubner, Stuttgart, 1977. [7] A. Ghose Choudhury and P. Guha, On isochronous cases of the Cherkas system and Jacobi's last multiplier, J. Phys. A: Math. Theor. 43 (2010), 125202, 12pp. [8] A. Ghose Choudhury and P. Guha, An analytic technique for the solutions of nonlinear oscillators with damping using the Abel Equation, to appear in Discontinuity, Nonlinearity and Complexity. [9] A. Ghose Choudhury, P. Guha and B. Khanra, On the Jacobi last multiplier, integrating factors and the Lagrangian formulation of differential equations of the Painlevé-Gambier classification, J. Math. Anal. Appl., 360 (2009), 651-664. [10] M. Grmela, Hamiltonian extended thermodynamics, J. Phys. A: Math. Gen., 23 (1990), 3341-3351. [11] P. Guha and A. Ghose Choudhury, The Jacobi last multiplier and isochronicity of Liénard type systems, Rev. Math. Phys., 25 (2013), 1330009, 31pp. [12] P. Guha, Metriplectic structure, Leibniz dynamics and dissipative systems, J. Math. Anal. Appl., 326 (2007), 121-136. [13] T. Harko, F. S. N. Lobo and M. K. Mak, A class of exact solutions of the Liénard type ordinary non-linear differential equation arXiv: 1302.0836v3[math-ph]. [14] T. Harko, F. S. N. Lobo and M. K. Mak, A Chiellini type integrability condition for the generalized first kind Abel differential equation, Universal Journal of Applied Mathematics, 1 (2013), 101-104. [15] C. Jacobi, Sul principio dellúltimo moltiplicatore, e suo uso come nuovo principio generale di meccanica, Giornale Arcadico di Scienze, Lettere ed Arti 99 (1844), 129-146. [16] C. Jacobi, A. Clebsch and C. Brockhardt, Jacobi's Lectures on Dynamics, Texts and Readings in Mathematics, Hindustan Book Agency, 2009. [17] A. N. Kaufman, Dissipative Hamiltonian systems: A unifying principle. Phys. Lett.A, 100 (1984), 419-422. [18] A. Lichnerowicz, Les variétés de Poisson et leurs algébres de Lie associées, J. Differential Geom., 12 (1977), 253-300. [19] S. C. Mancas and H. C. Rosu, Integrable dissipative nonlinear second order differential equations via factorizations and Abel equations, Phys. Lett. A, 377 (2013), 1234-1238. [20] S. C. Mancas and H. C. Rosu, Integrable equations with Ermakov-Pinney nonlinearities and Chiellini damping, Appl. Math. Comp., 259 (2015), 1-11. [21] M. K. Mak, H. W. Chan and T. Harko, Solutions generating technique for Abel-type nonlinear ordinary differential equations, Comput. Math. Appl., 41 (2001), 1395-1401. [22] P. J. Morrison, A paradigm for joined Hamiltonian and dissipative systems, Physica D, 18 (1986), 410-419. [23] P. J. Morrison, Thoughts on brackets and dissipation: Old and new, J. Phys.: Conf. Ser., 169 (2009), 012006. [24] M. C. Nucci and P. G. L. Leach, Jacobi's last multiplier and symmetries for the Kepler problem plus a lineal story, J. Phys. A: Math. Gen., 37 (2004), 7743-7753. [25] M. C. Nucci and P. G. L. Leach, The Jacobi's Last Multiplier and its applications in mechanics, Phys. Scr., 78 (2008), 065011. [26] M. C. Nucci and K. M. Tamizhmani, Lagrangians for dissipative nonlinear oscillators: The method of Jacobi last multiplier, Journal of Nonlinear Mathematical Physics, 17 (2010), 167-178. [27] S. G. Rajeev, A canonical formulation of dissipative mechanics using complex-valued Hamiltonians, Ann. Physics, 322 (2007), 1541-1555. [28] B. S. Madhava Rao, On the reduction of dynamical equations to the Lagrangian form, Proc. Benaras Math. Soc. (N.S.), 2 (1940), 53-59. [29] A. Raouf Chouikha, Isochronous centers of Lienard type equations and applications, J. Math. Anal. Appl., 331 (2007), 358-376. [30] H. C. Rosu, S. C. Mancas and P. Chen, Barotropic FRW cosmolog ies with Chiellini damping, Phys. Lett. A, 379 (2015), 882-887. [31] M. Sabatini, On the period Function of Liénard Systems, J. Diff. Eqns., 152 (1999), 467-487. [32] T. Shah, R. Chattopadhyay, K. Vaidya and Sagar Chakraborty, Conservative perturbation theory for nonconservative systems, Phys. Rev. E, 92 (2015), 062927. [33] I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Birkhäuser-Verlag, Basel, 1994. [34] E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge Mathematical Library. Cambridge University Press,, Cambridge, 1988.
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