Group Analysis of Non-autonomous Linear Hamiltonians through Differential Galois Theory

TL;DR

This paper introduces a new notion of integrability for non-autonomous Hamiltonian systems and establishes its equivalence to classical integrability in specific cases, providing classifications and reciprocal results.

Contribution

It defines a non-autonomous integrability concept, proves its equivalence to classical integrability for certain Hamiltonians, and classifies systems via symplectic transformations involving algebraic time functions.

Findings

Non-autonomous integrability is equivalent to classical integrability in specified cases.

Provides a reciprocal of Morales-Ramis result for quadratic homogeneous Hamiltonians.

Classifies systems by symplectic transformations with algebraic functions of time.

Abstract

In this paper we introduce a notion of integrability in the non autonomous sense. For the cases of 1 + 1/2 degrees of freedom and quadratic homogeneous Hamiltonians of 2 + 1/2 degrees of freedom we prove that this notion is equivalent to the classical complete integrability of the system in the extended phase space. For the case of quadratic homogeneous Hamiltonians of 2 + 1/2 degrees of freedom we also give a reciprocal of the Morales-Ramis result. We classify those systems by terms of symplectic change of frames involving algebraic functions of time, and give their canonical forms.