Note on integrability of certain homogeneous Hamiltonian systems

TL;DR

This paper studies a class of homogeneous Hamiltonian systems with two degrees of freedom, deriving necessary integrability conditions using differential Galois theory and particular solutions.

Contribution

It introduces a method to analyze integrability of coordinate-dependent kinetic energy Hamiltonian systems via differential Galois groups.

Findings

Derived necessary conditions for integrability.

Identified particular solutions based on homogeneity.

Applied differential Galois theory to variational equations.

Abstract

In this paper we investigate a class of natural Hamiltonian systems with two degrees of freedom. The kinetic energy depends on coordinates but the system is homogeneous. Thanks to this property it admits, in a general case, a particular solution. Using this solution we derive necessary conditions for the integrability of such systems investigating differential Galois group of variational equations.