Non-standard 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, $\ddot{x}+αx\dot{x}+βx^3+γx=0,$ which preserves the form of the time independent integral, conservative Hamiltonian and the equation of motion. Generalizing this transformation we prove the existence of non-standard 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 $\ddot{x}+αx^q\dot{x}+βx^{2q+1}=0$, 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. The associated Lagrangian description for all the above systems is also briefly discussed.