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 & 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.   Google Scholar

[2]

A. M. BlochP. S. KrishnaprasadJ. E. Marsden and T. Ratiu, The Euler-Poincaré equations and double Bracket dissipation, Comm. Math. Phys., 175 (1996), 1-42.   Google Scholar

[3]

A. M. BlochP. 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.   Google Scholar

[4]

A. Chiellini, Sullíntegrazione della equazione differenziale y' + Py2 + Qy3 = 0, Bollettino della Unione Matem-atica Italiana, 10 (1931), 301-307.   Google Scholar

[5]

C. Christopher and J. Devlin, On the classification of Liénard systems with amplitudeindependent periods, J. Differential Equations, 200 (2004), 1-17.   Google Scholar

[6]

E. Kamke, Differentialgleichungen: Lösungsmethoden und Lösungen, Neunte Auflage. Mit einem Vorwort von Detlef Kamke. B. G. Teubner, Stuttgart, 1977. Google Scholar

[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.   Google Scholar

[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. Google Scholar

[9]

A. Ghose ChoudhuryP. 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.   Google Scholar

[10]

M. Grmela, Hamiltonian extended thermodynamics, J. Phys. A: Math. Gen., 23 (1990), 3341-3351.   Google Scholar

[11]

P. Guha and A. Ghose Choudhury, The Jacobi last multiplier and isochronicity of Liénard type systems, Rev. Math. Phys., 25 (2013), 1330009.   Google Scholar

[12]

P. Guha, Metriplectic structure, Leibniz dynamics and dissipative systems, J. Math. Anal. Appl., 326 (2007), 121-136.   Google Scholar

[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]. Google Scholar

[14]

T. HarkoF. 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.   Google Scholar

[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.   Google Scholar

[16]

C. Jacobi, A. Clebsch and C. Brockhardt, Jacobi's Lectures on Dynamics, Texts and Readings in Mathematics, Hindustan Book Agency, 2009. Google Scholar

[17]

A. N. Kaufman, Dissipative Hamiltonian systems: A unifying principle, Phys. Lett.A, 100 (1984), 419-422.   Google Scholar

[18]

A. Lichnerowicz, Les variétés de Poisson et leurs algébres de Lie associées, J. Differential Geom., 12 (1977), 253-300.   Google Scholar

[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.   Google Scholar

[20]

S. C. Mancas and H. C. Rosu, Integrable equations with Ermakov-Pinney nonlinearities and Chiellini damping, Appl. Math. Comp., 259 (2015), 1-11.   Google Scholar

[21]

M. K. MakH. W. Chan and T. Harko, Solutions generating technique for Abel-type nonlinear ordinary differential equations, Comput. Math. Appl., 41 (2001), 1395-1401.   Google Scholar

[22]

P. J. Morrison, A paradigm for joined Hamiltonian and dissipative systems, Physica D, 18 (1986), 410-419.   Google Scholar

[23]

P. J. Morrison, Thoughts on brackets and dissipation: Old and new, J. Phys.: Conf. Ser., 169 (2009), 012006.   Google Scholar

[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.   Google Scholar

[25]

M. C. Nucci and P. G. L. Leach, The Jacobi's Last Multiplier and its applications in mechanics, Phys. Scr., 78 (2008), 065011.   Google Scholar

[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.   Google Scholar

[27]

S. G. Rajeev, A canonical formulation of dissipative mechanics using complex-valued Hamiltonians, Ann. Physics, 322 (2007), 1541-1555.   Google Scholar

[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.   Google Scholar

[29]

A. Raouf Chouikha, Isochronous centers of Lienard type equations and applications, J. Math. Anal. Appl., 331 (2007), 358-376.   Google Scholar

[30]

H. C. RosuS. C. Mancas and P. Chen, Barotropic FRW cosmolog ies with Chiellini damping, Phys. Lett. A, 379 (2015), 882-887.   Google Scholar

[31]

M. Sabatini, On the period Function of Liénard Systems, J. Diff. Eqns., 152 (1999), 467-487.   Google Scholar

[32]

T. ShahR. ChattopadhyayK. Vaidya and Sagar Chakraborty, Conservative perturbation theory for nonconservative systems, Phys. Rev. E, 92 (2015), 062927.   Google Scholar

[33]

I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Birkhäuser-Verlag, Basel, 1994. Google Scholar

[34] E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge Mathematical Library. Cambridge University Press,, Cambridge, 1988.   Google Scholar

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.   Google Scholar

[2]

A. M. BlochP. S. KrishnaprasadJ. E. Marsden and T. Ratiu, The Euler-Poincaré equations and double Bracket dissipation, Comm. Math. Phys., 175 (1996), 1-42.   Google Scholar

[3]

A. M. BlochP. 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.   Google Scholar

[4]

A. Chiellini, Sullíntegrazione della equazione differenziale y' + Py2 + Qy3 = 0, Bollettino della Unione Matem-atica Italiana, 10 (1931), 301-307.   Google Scholar

[5]

C. Christopher and J. Devlin, On the classification of Liénard systems with amplitudeindependent periods, J. Differential Equations, 200 (2004), 1-17.   Google Scholar

[6]

E. Kamke, Differentialgleichungen: Lösungsmethoden und Lösungen, Neunte Auflage. Mit einem Vorwort von Detlef Kamke. B. G. Teubner, Stuttgart, 1977. Google Scholar

[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.   Google Scholar

[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. Google Scholar

[9]

A. Ghose ChoudhuryP. 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.   Google Scholar

[10]

M. Grmela, Hamiltonian extended thermodynamics, J. Phys. A: Math. Gen., 23 (1990), 3341-3351.   Google Scholar

[11]

P. Guha and A. Ghose Choudhury, The Jacobi last multiplier and isochronicity of Liénard type systems, Rev. Math. Phys., 25 (2013), 1330009.   Google Scholar

[12]

P. Guha, Metriplectic structure, Leibniz dynamics and dissipative systems, J. Math. Anal. Appl., 326 (2007), 121-136.   Google Scholar

[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]. Google Scholar

[14]

T. HarkoF. 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.   Google Scholar

[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.   Google Scholar

[16]

C. Jacobi, A. Clebsch and C. Brockhardt, Jacobi's Lectures on Dynamics, Texts and Readings in Mathematics, Hindustan Book Agency, 2009. Google Scholar

[17]

A. N. Kaufman, Dissipative Hamiltonian systems: A unifying principle, Phys. Lett.A, 100 (1984), 419-422.   Google Scholar

[18]

A. Lichnerowicz, Les variétés de Poisson et leurs algébres de Lie associées, J. Differential Geom., 12 (1977), 253-300.   Google Scholar

[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.   Google Scholar

[20]

S. C. Mancas and H. C. Rosu, Integrable equations with Ermakov-Pinney nonlinearities and Chiellini damping, Appl. Math. Comp., 259 (2015), 1-11.   Google Scholar

[21]

M. K. MakH. W. Chan and T. Harko, Solutions generating technique for Abel-type nonlinear ordinary differential equations, Comput. Math. Appl., 41 (2001), 1395-1401.   Google Scholar

[22]

P. J. Morrison, A paradigm for joined Hamiltonian and dissipative systems, Physica D, 18 (1986), 410-419.   Google Scholar

[23]

P. J. Morrison, Thoughts on brackets and dissipation: Old and new, J. Phys.: Conf. Ser., 169 (2009), 012006.   Google Scholar

[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.   Google Scholar

[25]

M. C. Nucci and P. G. L. Leach, The Jacobi's Last Multiplier and its applications in mechanics, Phys. Scr., 78 (2008), 065011.   Google Scholar

[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.   Google Scholar

[27]

S. G. Rajeev, A canonical formulation of dissipative mechanics using complex-valued Hamiltonians, Ann. Physics, 322 (2007), 1541-1555.   Google Scholar

[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.   Google Scholar

[29]

A. Raouf Chouikha, Isochronous centers of Lienard type equations and applications, J. Math. Anal. Appl., 331 (2007), 358-376.   Google Scholar

[30]

H. C. RosuS. C. Mancas and P. Chen, Barotropic FRW cosmolog ies with Chiellini damping, Phys. Lett. A, 379 (2015), 882-887.   Google Scholar

[31]

M. Sabatini, On the period Function of Liénard Systems, J. Diff. Eqns., 152 (1999), 467-487.   Google Scholar

[32]

T. ShahR. ChattopadhyayK. Vaidya and Sagar Chakraborty, Conservative perturbation theory for nonconservative systems, Phys. Rev. E, 92 (2015), 062927.   Google Scholar

[33]

I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Birkhäuser-Verlag, Basel, 1994. Google Scholar

[34] E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge Mathematical Library. Cambridge University Press,, Cambridge, 1988.   Google Scholar
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