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

TL;DR

This paper uncovers a nonlocal transformation linking damped harmonic oscillators to nonlinear equations, revealing non-standard Hamiltonian structures and solutions for various damped nonlinear systems.

Contribution

It introduces a novel nonlocal transformation that preserves Hamiltonian structures in damped nonlinear oscillators, extending to generalized Emden and Mathews-Lakshmanan systems.

Findings

Established a nonlocal transformation between damped harmonic and nonlinear oscillators.

Derived non-standard Hamiltonian structures for a class of damped nonlinear systems.

Obtained explicit solutions for the generalized modified Emden equation.

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, $\ddot{x}+\alpha x\dot{x}+\beta x^3+\gamma x=0,$ 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 Li\'enard type systems. Further, using the above Hamiltonian structure for a specific example namely the generalized modified Emden equation $\ddot{x}+\alpha x^q\dot{x}+\beta x^{2q+1}=0$, where $\alpha$, $\beta$ 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.