Integrability of the mixmaster model

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.