Configurational invariants of Hamiltonian systems

TL;DR

This paper investigates conditions under which 2D Hamiltonian systems have polynomial invariants, distinguishing between weak and strong integrability, and provides new examples of such systems.

Contribution

It characterizes the form of polynomial invariants for Hamiltonian systems and introduces new examples of weakly integrable systems with linear and quadratic invariants.

Findings

Leading order coefficient is an arbitrary holomorphic function in weak integrability.

Strong integrability requires the polynomial to be in the coordinates.

New examples of weakly integrable systems with linear and quadratic invariants.

Abstract

In this paper we explore the general conditions in order that a 2-dimensional natural Hamiltonian system possess a second invariant which is a polynomial in the momenta and is therefore Liouville integrable. We examine the possibility that the invariant is preserved by the Hamiltonian flow on a given energy hypersurface only (weak integrability) and derive the additional requirement necessary to have conservation at arbitrary energy (strong integrability). Using null complex coordinates, we show that the leading order coefficient of the polynomial is an arbitrary holomorphic function in the case of weak integrability and a polynomial in the coordinates in the strongly integrable one. We review the results obtained so far with strong invariants up to degree four and provide some new examples of weakly integrable systems with linear and quadratic invariants.