On the Integrability of Bianchi Cosmological Models

TL;DR

This paper investigates the mathematical integrability of Bianchi class A cosmological models, analyzing their Hamiltonian structure, dynamics, and integrability criteria to understand their evolution and potential for analytical solutions.

Contribution

It provides a detailed analysis of the integrability of Bianchi class A models, including reduction of phase space and proof of non-existence of first integrals.

Findings

Phase space can be reduced by two dimensions.

The vector field of the reduced system is polynomial.

No analytic or formal first integrals exist for the system.

Abstract

In this work, we are investigating the problem of integrability of Bianchi class A cosmological models. This class of systems is reduced to the form of Hamiltonian systems with exponential potential forms. The dynamics of Bianchi class A models is investigated through the Euler-Lagrange equations and geodesic equations in the Jacobi metric. On this basis, we have come to some general conclusions concerning the evolution of the volume function of 3-space of constant time. The formal and general form of this function has been found. It can serve as a controller during numerical calculations of the dynamics of cosmological models. The integrability of cosmological models is also discussed from the points of view of different integrability criterions. We show that dimension of phase space of Bianchi class A Hamiltonian systems can be reduced by two. We prove vector field of the reduced system is polynomial and it does not admit any analytic, or even formal first integral.