Finding the Hannay angle in dissipative oscillatory systems via conservative perturbation theory.

@article{Chattopadhyay2018FindingTH,
  title={Finding the Hannay angle in dissipative oscillatory systems via conservative perturbation theory.},
  author={Rohitashwa Chattopadhyay and Tirth Shah and Sagar Chakraborty},
  journal={Physical review. E},
  year={2018},
  volume={97 6-1},
  pages={
          062209
        }
}
Usage of a Hamiltonian perturbation theory for a nonconservative system is counterintuitive and, in general, a technical impossibility by definition. However, the time-independent dual Hamiltonian formalism for the nonconservative systems has opened the door for using various conservative perturbation theories for investigating the dynamics of such systems. Here we demonstrate that the Lie transform Hamiltonian perturbation theory can be adapted to find the perturbative solutions and the… 
3 Citations

Figures from this paper

Complex dynamical properties of coupled Van der Pol-Duffing oscillators with balanced loss and gain
We consider a Hamiltonian system of coupled Van der Pol-Duffing(VdPD) oscillators with balanced loss and gain. The system is analyzed perturbatively by using Renormalization Group(RG) techniques as
Relaxation oscillations and frequency entrainment in quantum mechanics.
TLDR
The previously known steady state of such quantum oscillators in the weakly nonlinear regime is shown to emerge as a special case and the hallmark of strong nonlinearity-relaxation oscillations-is shown in quantum mechanics.
Classical analog of the quantum metric tensor.
We present a classical analog of the quantum metric tensor, which is defined for classical integrable systems that undergo an adiabatic evolution governed by slowly varying parameters. This classical

References

SHOWING 1-10 OF 116 REFERENCES
Conservative perturbation theory for nonconservative systems.
TLDR
This work surmounts the hitherto perceived barrier for canonical perturbation theory that it can be applied only to a class of conservative systems, viz., Hamiltonian systems, and finds Hamiltonian structure for an important subset of Liénard system-a paradigmatic system for modeling isolated and asymptotic oscillatory state.
Lie series method for vector fields and Hamiltonian perturbation theory
We consider a rigorous Hamiltonian perturbation theory based on the transformation of the vector field of the system, realized by the Lie method. Such a perturbative technique presents some
An extended canonical perturbation method
In this investigation, a procedure is described for extending the application of canonical perturbation theories, which have been applied previously to the study of conservative systems only, to the
Equivalent linearization finds nonzero frequency corrections beyond first order
Abstract We show that the equivalent linearization technique, when used properly, enables us to calculate frequency corrections of weakly nonlinear oscillators beyond the first order in nonlinearity.
Geometric phases in dissipative systems.
It is shown that a phenomenon analogous to the geometric phase shifts of Berry and Hannay occurs for dissipative oscillatory systems and can be detected in numerical simulations of chemical
Classical mechanics of nonconservative systems.
TLDR
A formulation of Hamilton's principle that is compatible with initial value problems is presented, which leads to a natural formulation for the Lagrangian and Hamiltonian dynamics of generic nonconservative systems, thereby filling a long-standing gap in classical mechanics.
Unusual Liénard-type nonlinear oscillator.
A Liénard type nonlinear oscillator of the form x+kxx+(k2/9)x3+lambda1x=0, which may also be considered as a generalized Emden-type equation, is shown to possess unusual nonlinear dynamical
On the Lagrangian and Hamiltonian description of the damped linear harmonic oscillator
Using the modified Prelle-Singer approach, we point out that explicit time independent first integrals can be identified for the damped linear harmonic oscillator in different parameter regimes.
Angle variable holonomy in adiabatic excursion of an integrable Hamiltonian
If an integrable classical Hamiltonian H describing bound motion depends on parameters which are changed very slowly then the adiabatic theorem states that the action variables I of the motion are
...
...