Integrability of the mixmaster model

TL;DR

This paper demonstrates that the anisotropic Bianchi type IX mixmaster model in vacuum is locally Liouville integrable by identifying nonlocal conserved charges and exploiting parametrization invariance, supporting previous findings.

Contribution

It introduces a generalized definition of nonlocal conserved charges and shows their role in establishing the integrability of the mixmaster model.

Findings

The model is locally Liouville integrable.

Existence of two sets of four independent conserved charges.

Conserved charges depend explicitly on time due to parametrization invariance.

Abstract

The mixmaster model has always been a field of controversy in the literature regarding its (non)integrability. In this work, we make use of a generalized definition of a class of nonlocal conserved charges in phase space to demonstrate that the anisotropic Bianchi type IX model in vacuum is -at least locally - Liouville integrable, thus supporting the findings of previous works pointing to this result. These additional integrals of motion that we use can be defined only due to the parametrization invariance of the system and can be seen to possess an explicit dependence on time. By promoting the time variable to a degree of freedom, we demonstrate the existence of two sets of four independent conserved charges that are in involution, thus leading to the characterization of the system as integrable in terms of the Liouville-Arnold theorem.