Hannay angle and geometric phase shifts under adiabatic parameter changes in classical dissipative systems

TL;DR

This paper links the classical Hannay angle to geometric phase shifts in dissipative systems, suggesting a broader applicability including stochastic dynamics, thus extending the understanding of geometric phases in classical mechanics.

Contribution

It demonstrates that the geometric phase in dissipative limit cycles is equivalent to the classical Hannay angle, and proposes its generalization to stochastic systems.

Findings

Geometric phase in dissipative systems equals the Hannay angle.

The phase concept extends to stochastic evolution with noise.

Provides a unified view of geometric phases in classical mechanics.

Abstract

In Phys. Rev. Lett. {\bf 66}, 847 (1991), T. B. Kepler and M. L. Kagan derived a geometric phase shift in dissipative limit cycle evolution. This effect was considered as an extension of the geometric phase in classical mechanics. We show that the converse is also true, namely, this geometric phase can be identified with the classical mechanical Hannay angle in an extended phase space. Our results suggest that this phase can be generalized to a stochastic evolution with an additional noise term in evolution equations.