Phase Splitting for Periodic Lie Systems

TL;DR

This paper introduces a phase splitting method for periodic Lie systems within Floquet theory, decomposing the monodromy logarithm into dynamic and geometric components with geometric interpretation.

Contribution

It presents a novel phase splitting technique for real periodic Lie systems, linking dynamic phase to Hamiltonian dynamics and geometric phase to symplectic geometry.

Findings

Dynamic phase is intrinsic and related to the Hamiltonian.

Geometric phase is expressed as a surface integral of the symplectic form.

The method provides a new perspective on the structure of monodromy in Lie systems.

Abstract

In the context of the Floquet theory, using a variation of parameter argument, we show that the logarithm of the monodromy of a real periodic Lie system with appropriate properties admits a splitting into two parts, called dynamic and geometric phases. The dynamic phase is intrinsic and linked to the Hamiltonian of a periodic linear Euler system on the co-algebra. The geometric phase is represented as a surface integral of the symplectic form of a co-adjoint orbit.