Integrability of certain Hamiltonian systems in $2D$ variable curvature spaces

TL;DR

This paper investigates the integrability of Hamiltonian systems in two-dimensional variable curvature spaces using differential Galois theory, providing necessary conditions and new integrable examples to advance understanding of nonlinear dynamics in curved geometries.

Contribution

It introduces a differential Galois-based method to determine integrability conditions for Hamiltonian systems in variable curvature spaces, including new examples.

Findings

Necessary integrability conditions in terms of parameter restrictions

Application of results to specific examples demonstrating effectiveness

Discovery of new integrable Hamiltonian systems in curved spaces

Abstract

The objective of this work is to examine the integrability of Hamiltonian systems in $2D$ spaces with variable curvature of certain types. Based on the differential Galois theory, we announce the necessary conditions of the integrability. They are given in terms of arithmetic restrictions on values of the parameters describing the system. We apply the obtained results to some examples to illustrate that the applicability of the obtained result is easy and effective. Certain new integrable examples are given. The findings highlight the applicability of the differential Galois approach in studying the integrability of Hamiltonian systems in curved spaces, expanding our understanding of nonlinear dynamics and its potential applications.