Nonstandard conserved Hamiltonian structures in dissipative/damped systems : Nonlinear generalizations of damped harmonic oscillator

Abstract. In this paper we point out the existence of a remarkable nonlocal transformation between the damped harmonic oscillator and a modified Emden type nonlinear oscillator equation with linear forcing, ẍ+αxẋ+βx+γx = 0,which preserves the form of the time independent integral, conservative Hamiltonian and the equation of motion. Generalizing this transformation we prove the existence of nonstandard conservative Hamiltonian structure for a general class of damped nonlinear oscillators including Liénard type systems. Further, using the above Hamiltonian structure for a specific example namely the generalized modified Emden equation ẍ+αxẋ+βx = 0, where α, β and q are arbitrary parameters, the general solution is obtained through appropriate canonical transformations. We also present the conservative Hamiltonian structure of the damped Mathews-Lakshmanan oscillator equation.