Non-standard conserved Hamiltonian structures in dissipative/damped systems : Nonlinear generalizations of damped harmonic oscillator

2008/10/10 by R Gladwin Pradeep, V K Chandrasekar, Pradeep, R Gladwin +5
Physics and Astronomy · #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #nlin.SI

paper · pdf · doi:10.48550/arxiv.0810.1819

Accepted for publication in J. Math. Phys

arxiv created 2009/04/13 · arxiv updated 2009/12/01

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, x+αxx+βx³+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 x+αx^qx+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.

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