On the integrability of the Hamiltonian systems with homogeneous polynomial potentials

Jaume Llibre,Xiang Zhang

doi:10.2478/amns.2018.2.00041

Abstract

Abstract We summarize the known results on the integrability of the complex Hamiltonian systems with two degrees of freedom defined by the Hamiltonian functions of the form H = 1 2 ∑ i = 1 2 p i 2 + V ( q 1 , q 2 ) , $$\begin{array}{} \displaystyle H=\frac{1}{2}\sum_{i=1}^{2}p_i^2+V(q_1,q_2), \end{array} $$ where V(q 1,q 2) are homogeneous polynomial potentials of degree k.

Highlights

In the theory of ordinary differential equations and in particular in the theory of Hamiltonian systems the existence of first integrals is important, because they allow to lower the dimension where the Hamiltonian system is defined
If we know a sufficient number of first integrals, these allow to solve the Hamiltonian system explicitly, and we say that the system is integrable
Almost until the end of the 19th century the major part of mathematicians and physicians believe that the equations of classical mechanics were integrable, and that to find their first integrals was mainly a computational problem

Summary

Introduction

In the theory of ordinary differential equations and in particular in the theory of Hamiltonian systems the existence of first integrals is important, because they allow to lower the dimension where the Hamiltonian system is defined. First we summarize the classification of all complex Hamiltonian systems (1) with homogeneous polynomial potentials of degree k ∈ {−2, −1, 0, 1, 2, 3, 4}, which are integrable with meromorphic first integrals. As we shall see for all these Hamiltonian systems, except for the ones with potential of degree −2, the meromorphic first integral independent of the Hamiltonian can be chosen polynomial. We summarize the results on the Hamiltonian systems (1) with homogeneous polynomial potentials of degree −3 which are integrable with polynomial first integrals. We present the results on the integrability of the Hamiltonian systems (1) with the so called exceptional homogeneous polynomial potentials of degree k > 4. As far as we know at this moment it is an open question to provide a complex Hamiltonian system (1) with a homogeneous polynomial potential of degree k > 0 which is integrable with meromorphic first integrals, and such that it has no polynomial first integrals independent of the Hamiltonian

Equivalent potentials

Morales–Ruiz and Ramis results

Homogeneous polynomial potentials of degree 3

Homogeneous polynomial potentials of degree 4

The exceptional homogeneous polynomial potentials