Note on integrability of certain homogeneous Hamiltonian systems in 2D constant curvature spaces

TL;DR

This paper establishes necessary conditions for the integrability of specific Hamiltonian systems in 2D constant curvature spaces, extending the concept of homogeneous potentials to curved geometries and analyzing their differential Galois groups.

Contribution

It introduces a framework for analyzing integrability of Hamiltonian systems with potentials analogous to homogeneous potentials in curved spaces, using differential Galois theory.

Findings

Derived necessary integrability conditions for these systems.

Identified the role of differential Galois groups in integrability.

Provided a method to analyze Hamiltonian systems in curved geometries.

Abstract

We formulate the necessary conditions for the integrability of a certain family of Hamiltonian systems defined in the constant curvature two-dimensional spaces. Proposed form of potential can be considered as a counterpart of a homogeneous potential in flat spaces. Thanks to this property Hamilton equations admit, in a general case, a particular solution. Using this solution we derive necessary integrability conditions investigating differential Galois group of variational equations.