Nonstandard conserved Hamiltonian structuresin dissipative/damped systems: Nonlineargeneralizations of damped harmonic oscillator

摘要

In this paper we point out the existence of a remarkable nonlocal transformationbetween the damped harmonic oscillator and a modified Emden-type nonlinearoscillator equation with linear forcing,x+αxx+βx~3+ γx=0,which preserves theform of the time independent integral, conservative Hamiltonian, and the equationof motion. Generalizing this transformation we prove the existence of nonstandardconservative Hamiltonian structure for a general class of damped nonlinear oscil-lators including Lienard-type systems. Further, using the above Hamiltonian struc-ture for a specific example, namely, the generalized modified Emden equation αx~qx+βx~(2q+)l=0,wherea, 0,andqare arbitrary parameters, the general solutionis obtained through appropriate canonical transformations. We also present theconservative Hamiltonian structure of the damped Mathews–Lakshmanan oscillatorequation. The associated Lagrangian description for all the above systems is alsobriefly discussed.