Differential Galois theory and Integrability

📝 Abstract

This paper is an overview of our works which are related to investigations of the integrability of natural Hamiltonian systems with homogeneous potentials and Newton’s equations with homogeneous velocity independent forces. The two types of integrability obstructions for these systems are presented. The first, local ones, are related to the analysis of the differential Galois group of variational equations along a non-equilibrium particular solution. The second, global ones, are obtained from the simultaneous analysis of variational equations related to all particular solutions belonging to a certain class. The marriage of these two types of the integrability obstructions enables to realise the classification programme of all integrable homogeneous systems. The main steps of the integrability analysis for systems with two and more degrees of freedom as well as new integrable systems are shown.

💡 Analysis

This paper is an overview of our works which are related to investigations of the integrability of natural Hamiltonian systems with homogeneous potentials and Newton’s equations with homogeneous velocity independent forces. The two types of integrability obstructions for these systems are presented. The first, local ones, are related to the analysis of the differential Galois group of variational equations along a non-equilibrium particular solution. The second, global ones, are obtained from the simultaneous analysis of variational equations related to all particular solutions belonging to a certain class. The marriage of these two types of the integrability obstructions enables to realise the classification programme of all integrable homogeneous systems. The main steps of the integrability analysis for systems with two and more degrees of freedom as well as new integrable systems are shown.

📄 Content

arXiv:0912.1046v1 [nlin.SI] 5 Dec 2009 Differential Galois theory and Integrability Andrzej J. Maciejewski1 and Maria Przybylska2 1J. Kepler Institute of Astronomy, University of Zielona G´ora, Licealna 9, PL-65–417 Zielona G´ora, Poland (e-mail: maciejka@astro.ia.uz.zgora.pl) 2Toru´n Centre for Astronomy, N. Copernicus University, Gagarina 11, PL-87–100 Toru´n, Poland, (e-mail:Maria.Przybylska@astri.uni.torun.pl) Abstract This paper is an overview of our works which are related to investigations of the integrability of natural Hamiltonian systems with homogeneous potentials and New- ton’s equations with homogeneous velocity independent forces. The two types of in- tegrability obstructions for these systems are presented. The first, local ones, are re- lated to the analysis of the differential Galois group of variational equations along a non-equilibrium particular solution. The second, global ones, are obtained from the simultaneous analysis of variational equations related to all particular solutions belong- ing to a certain class. The marriage of these two types of the integrability obstructions enables to realise the classification programme of all integrable homogeneous systems. The main steps of the integrability analysis for systems with two and more degrees of freedom as well as new integrable systems are shown. keywords:integrability; differential Galois theory; Hamiltonian systems. 1 Introduction It is hard to imagine yourself a world without integrable models. Teaching in such a world would be rather frustrating. If a theory has no solvable examples, it is difficult to explain that it is useful. Fortunately, in ours we have the harmonic oscillator—the guinea-pig which serves as pedagogical example for all theories. Integrable models are exceptional, but we do not neglect them. Still, as it was one century ago, finding a new non-trivially integrable model is a discovery. How we can find integrable systems? One way, which seems to be the most natural, is just a search them in the nature. That is, start with more or less general model, and look for some special cases when the model is integrable. The other way is to construct integrable systems. It appears that the first approach is much more difficult than the second one. The reason of this is obvious: we know only few general methods which give strongly enough, and computable necessary conditions for the integrability. In this paper we consider only classical dynamical systems which are described by or- dinary differential equations. There is no a unique definition of such systems and there is no a unique method for study of their integrability. Nevertheless, in this paper we will con- centrate mainly on applications of only one quite new theory. It was developed by Baider, Churchill, Morales, Ramis, Rod, Sim´o and Singer in the end of the XX century, see [3; 5; 32] and references therein. In the context of Hamiltonian systems it is called the Morales-Ramis theory. It arose from very long searching for relations between the branching of solutions of differential equations considered as functions of complex time, and the integrability. In the context of Hamiltonian systems these relations were found by S. L. Ziglin [50; 51]. His elegant theory expresses necessary conditions for the integrability in terms of properties of the monodromy group of variational equations along a particular solution. The Morales- Ramis theory, in some sense, is an algebraic version of the Ziglin theory. It formulates the necessary conditions for the integrability in terms of the differential Galois group of the variational equations. During one and half decade the Morales-Ramis theory was applied successfully for study the integrability of numerous systems, see an overview paper [35]. Let us mention two biggest successes of this theory. It was applied to prove the non-integrability of the planar three body problem [12; 47; 48; 49], and to prove the non-integrability of the Hill lunar problem [37]. Moreover, as it is well known, the first proof of the fact that the problem of the heavy top is integrable only in the classical cases was done by S. L. Ziglin in [51] and it is based on his theory. An alternative proof based on differential Galois approach is given in [28]. The above examples show the real power of the differential Galois methods in a study of the integrability. During last ten years we applied them to study several Hamiltonian and non-Hamiltonian systems which appear in physics and astronomy, see, e.g., [6; 20; 21; 22; 23; 25; 26; 29; 30; 31; 44]. Always a hidden dream of those investigations was a strong will to find an unexpectedly integrable system. However, for a long time, neither we, nor other authors succeeded with this respect. All those successful investigations gave negative results: the investigated systems are not integrable except already known integrable cases. In this paper our aim is to give an overview of our works concerning the integrability of Hami

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