Non-integrability criterion for homogeneous Hamiltonian systems via blowing-up technique of singularities

TL;DR

This paper introduces a new method using blowing-up theory to determine non-integrability in homogeneous Hamiltonian systems with two degrees of freedom, expanding the tools available for analyzing complex dynamical systems.

Contribution

It presents a novel approach based on blowing-up singularities to prove non-integrability, applicable to systems with non-integer homogeneous degrees.

Findings

Provides a new non-integrability criterion for homogeneous Hamiltonian systems.

Extends the blowing-up technique to systems with real homogeneous degrees.

Offers a comparison with the Molares-Ramis theory, highlighting its effectiveness.

Abstract

It is a big problem to distinguish between integrable and non-integrable Hamiltonian systems. We provide a new approach to prove the non-integrability of homogeneous Hamiltonian systems with two degrees of freedom. The homogeneous degree can be chosen from real values (not necessarily integer). The proof is based on the blowing-up theory which McGehee established in the collinear three-body problem. We also compare our result with Molares-Ramis theory which is the strongest theory in this field.