Integrable magnetic geodesic flows on Lie groups

TL;DR

This paper investigates magnetic geodesic flows on Lie groups, providing conditions for integrability, classifying 2-cocycles, and identifying integrable cases among 4-dimensional Lie algebras.

Contribution

It introduces a comprehensive framework for analyzing integrability of magnetic geodesic flows on Lie groups, including classification of 2-cocycles and explicit integrability criteria.

Findings

Necessary and sufficient conditions for integrability in quadratures.

Canonical forms for 2-cocycles of 4-dimensional Lie algebras.

Identification of integrable cases among 4-dimensional Lie algebras.

Abstract

Right-invariant geodesic flows on manifolds of Lie groups associated with 2-cocycles of corresponding Lie algebras are discussed. Algebra of integrals of motion for magnetic geodesic flows is considered and necessary and sufficient condition of integrability in quadratures is formulated. Canonic forms for 2-cocycles of all 4-dimensional Lie algebras are given and integrable cases among them are separated.