nonintegrability of bianchi viii hamiltonian system

Nonintegrability of Bianchi VIII Hamiltonian systemAndrzej J. Maciejewski, Jean-Marie Strelcyn, and Marek Szydłowski Citation: Journal of Mathematical Physics 42, 1728 (2001); doi: 10.1063/1.1351885 View online: http://dx.doi.org/10.1063/1.1351885 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/42/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effective action for noncommutative Bianchi I model AIP Conf. Proc. 1540, 113 (2013); 10.1063/1.4810779 Global dynamics for a relativistic charged matter in the presence of a massive scalar field and the presence of acosmological constant on Bianchi spacetimes J. Math. Phys. 53, 102502 (2012); 10.1063/1.4748310 The Hamiltonian formulation for the dynamics of a multishell self-gravitating system J. Math. Phys. 51, 072504 (2010); 10.1063/1.3431030 Equivalence classes of perturbations in cosmologies of Bianchi types I and V: Propagation and constraintequations J. Math. Phys. 41, 6890 (2000); 10.1063/1.1290378 The kinematical role of automorphisms in the orthonormal frame approach to Bianchi cosmology J. Math. Phys. 40, 353 (1999); 10.1063/1.532776

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.192.114.19 On: Thu, 18 Dec 2014 04:53:33

http://scitation.aip.org/content/aip/journal/jmp?ver=pdfcov

http://oasc12039.247realmedia.com/RealMedia/ads/click_lx.ads/www.aip.org/pt/adcenter/pdfcover_test/L-37/578858469/x01/AIP-PT/CiSE_JMPArticleDL_121714/Awareness_LibraryF.jpg/47344656396c504a5a37344142416b75?x

http://scitation.aip.org/search?value1=Andrzej+J.+Maciejewski&option1=author

http://scitation.aip.org/search?value1=Jean-Marie+Strelcyn&option1=author

http://scitation.aip.org/search?value1=Marek+Szyd�owski&option1=author

http://scitation.aip.org/content/aip/journal/jmp?ver=pdfcov

http://dx.doi.org/10.1063/1.1351885

http://scitation.aip.org/content/aip/journal/jmp/42/4?ver=pdfcov

http://scitation.aip.org/content/aip?ver=pdfcov

http://scitation.aip.org/content/aip/proceeding/aipcp/10.1063/1.4810779?ver=pdfcov

http://scitation.aip.org/content/aip/journal/jmp/53/10/10.1063/1.4748310?ver=pdfcov






Nonintegrability of Bianchi VIII Hamiltonian systemAndrzej J. Maciejewskia)

Torun Centre for Astronomy, N. Copernicus University,87–100 Torun, Gagarina 11, Poland and J. Kepler Astronomical Center, PedagogicalUniversity, Lubuska 2, 65-265 Zielona Go`ra, Poland

Jean-Marie Strelcynb)

Departement de Mathe´matiques, Universite´ de Rouen, UMR-CNRS 60-85, F 76821 MontSaint Aignan Cedex, France

Marek Szydłowskic)

Astronomical Observatory, Jagiellonian University, Orla 171, 30-244 Krako´w, Poland

~Received 27 July 2000; accepted for publication 13 December 2000!

In this paper we study the Bianchi VIII cosmological dynamical system. Our aim isto show that this system is nonintegrable. To show this we use an extension ofZiglin theory made by Morales-Ruiz and Ramis. ©2001 American Institute ofPhysics. @DOI: 10.1063/1.1351885#

I. INTRODUCTION

In certain cases cosmological models and relativistic systems can be represented as finitedimensional dynamical systems, i.e., as a set of ordinary differential equations. Such representa-tion must be considered a great advantage. Usually the original model is represented as a coupledsystem of partial nonlinear differential equations arising from the field theory. Having a finitedimensional dynamical system, we have at hand a very rich theory and elaborated tools for itsinvestigations.

It has to be mentioned, however, that dynamical systems of cosmological~or relativistic!origin have many special features which distinguish them from a ‘‘typical’’ dynamical system wemeet in dynamical astronomy, classical mechanics or physics~see Ref. 1!. These differences leadsometimes to controversies and long discussions. A good example is connected with the BianchiIX @B~IX ! or the Mixmaster# model and the presence of chaos in it. The numerically computedmaximal Lapunov exponent for this system was equal to zero or different from zero depending onthe time parametrization. Because an approximation of the Mixmaster model by a discrete maphas strong chaotic properties,2 it was natural to expect such behavior in the original system.However, as it was proved,3 there exists no recurrence in the system and, thus, no form of standarddeterministic chaos is present in it. Nevertheless, there were attempts to show that the dynamics ofB~IX ! contains some features characteristic for chaotic behavior. For example, in Refs. 4 and 5one can find results of interesting numerical works where ‘‘parametrization independent’’ charac-teristics of chaos were used. Let us note, however, that this approach does not make big progressin understanding the system. The point is that in the cited works a certain approximation of theMixmaster model was investigated, not the Mixmaster model itself. Moreover, one notices severalnotions used in these investigations which have no precise meaning.

A strict proof of the specific character of chaos in the Mixmaster system seems to be verydifficult. That is why several authors tried to show that this system is not integrable. It must bementioned that ‘‘integrability’’ here was understood differently by different authors. Several au-thors tested if the model passes the standard Painleve´ integrability test in the form of the ARS

a!Electronic mail: [email protected]!Also at Laboratoire Analyse, Ge´ometrie et Applications, UMR CNRS 7539, Institut Galile´e, Departement de Mathe´ma-

tiques, Universite´ Paris-Nord, Avenue J.-B. Cle´ment, 93430 Villetaneuse, France: Electronic mail: [email protected]; [email protected]

c!Electronic mail: [email protected]

JOURNAL OF MATHEMATICAL PHYSICS VOLUME 42, NUMBER 4 APRIL 2001

17280022-2488/2001/42(4)/1728/16/$18.00 © 2001 American Institute of Physics


128.192.114.19 On: Thu, 18 Dec 2014 04:53:33

algorithm.6 In this approach the Mixmaster system is extended to a complex dynamical system. InRefs. 7 and 8 it was indicated that B~IX ! model passes this test. This result was revised,9 however,without any strict conclusions concerning integrability. Further studies10 showed that the B~IX !model does not pass the so-called perturbative Painleve´ test. The authors of this article suggest theexistence of ‘‘some chaotic regimes’’ in the system, but, as the result of Cushman and S´niatyckishows, this claim without explanation of what ‘‘chaotic regimes’’ means is meaningless. Thestrongest result to this end was obtained in Ref. 11 where the authors show the existence ofmovable critical essential singularities in the B~IX ! model. This kind of investigation connects acomplicated behavior of the system with singularities of its solutions on the complex time plane.It should be mentioned, however, that the relation between Painleve´’s test and integrability, inparticular with complete integrability of Hamiltonian systems, is far from being clear.

The strongest and mathematically precise result concerning the Mixmaster model was ob-tained by J. J. Morales-Ruiz and J.-P. Ramis on the basis of their theory connecting Ziglin’smethod and differential Galois theory.12 We state this result briefly. The Mixmaster model can beformulated as a Hamiltonian system. There exists a four dimensional invariant manifoldT onwhich the system can be integrated explicitly. Solutions of the system restricted toT are known asTaub solutions. Studying variational equations around a selected Taub solution J. J. Morales-Ruizproved that the Mixmaster model, considered as a complex Hamiltonian system, is not completelyintegrable~with meromorphic first integrals! in the sense of Liouville–Arnold theorem. As thesystem is homogeneous the meromorphic first integrals can be replaced by rational first integrals.

We make a few remarks about these results. First, it was not excluded that this systempossesses one additional rational integral or is completely integrable in terms of nonrationalintegrals. Moreover, it can possess an additional integral only on the physically interesting zeroenergy level, or it can be integrable on it. Thus, by no means, does the result of Morales-Ruiz andRamis close the subject.

In the sequel, when speaking about a completely integrable Hamiltonian system, we assumethat all integrals are meromorphic.

In this article we apply Morales-Ruiz and Ramis approach to prove the nonintegrability of theBianchi VIII model. To this end we study its dynamics carefully. We show that this systemcontains a four dimensional subsystem which is completely integrable. We demonstrate how tofind an explicit form of solutions of this subsystem. Next, we consider particular solutions lying onthe zero level of the Hamiltonian, and we investigate the normal variational equations for thesesolutions. Using the Kovacic algorithm, we determine the differential Galois group of these equa-tions, and we prove the nonintegrability of the Bianchi VIII model.

II. DIFFERENTIAL GALOIS EXTENSION OF ZIGLIN THEORY

The fundamental papers of Ziglin13,14 gave a formulation of a very basic theorem about thenonintegrability of analytic Hamiltonian systems. The idea of the Ziglin approach lies in a deepconnection between properties of solutions on the complex time plane and the existence of firstintegral. This idea takes its origins from works of S. W. Kovalevskaya and A. M. Lapunov. Ziglinworks found many continuations and many important applications~see, e.g., Refs. 15–27!.

Here we give only the definitions needed for a formulation of the theorem we used to proveour main result.

We consider a complex Hamiltonian system withn degrees of freedom. It is defined on acomplex analytic symplectic manifoldM of complex dimension 2n, and is given by a holomor-phic Hamiltonian H:M→C. The Hamiltonian vector field associated with H is denoted byXH .

Let us assume that we know a nonequilibrium solutionz5w(t), tPC, of Hamilton’s equa-tions of motion generated byXH which is the maximal analytical continuation of a certain localsolution. We associate a Riemannian surfaceG with it. Next, we consider the variational equationsalongw(t), and we restrict them to the normal bundle ofG. These reduced variational equationsare called the normal variational equations~NVE!.

In the Ziglin approach the fundamental role is played by the monodromy group of NVE.Continuation along a close pathg with the base pointt0PC of a local fundamental solution of

1729J. Math. Phys., Vol. 42, No. 4, April 2001 Nonintegrability of B(VIII) Hamiltonian system


128.192.114.19 On: Thu, 18 Dec 2014 04:53:33

NVE gives rise to a new fundamental solution of NVE defined in the same neighborhood oft0 . AsNVE are linear and our system is Hamiltonian, there exists an elementGg of symplectic groupSp(2(n21),C) which transforms one solution to the other. In fact,Gg depends on the homotopyclass ofg, and mapg→Gg gives a representation of the first homotopy groupp1(G) of G inSp(2(n21),C). The image of this representation is called the monodromy group associated withNVE.

The Ziglin theory bases on two implications:

~i! If the considered Hamiltonian system possesses a first integral, then NVE also possess afirst integral.

~ii ! If NVE possess a first integral, then there exists a nonconstant function invariant withrespect to the action of the monodromy group.

Now, it is clear that the existence of first integrals puts certain constraints onto the monodromygroup and this fact was used by Ziglin to formulate his theorems.

J. J. Morales-Ruiz and J. P. Ramis modify the Ziglin approach in two respects. First, insteadof answering the question of what restrictions to the monodromy group are imposed by theexistence of certain numbers of independent first integrals, they ask what the consequences ofcomplete integrability are. Next, instead of working with the monodromy group of NVE, they usethe differential Galois group associated with NVE. This last group is bigger than the monodromygroup and this fact makes a proof of complete nonintegrability easier. For a definition of differ-ential Galois group, Liouvillian solutions and differential Galois theory~see Refs. 28–32!.

We prove the main result of this article using the following theorem.Theorem 1 „Morales-Ruiz, 1999…: Assume that there exist n first integrals of XH which are

meromorphic, in involution and functionally independent in a neighborhood ofG. Then the iden-tity component of the differential Galois group of NVE is Abelian.

The main difficulty in application of the above theorem is connected with determination of theidentity component of the differential Galois group of NVE. In fact, little is known about how todetermine it for given equations. However, in our case, as it will be shown, we can reduce theinvestigated NVE to a second order linear equation with rational coefficients. This allows us toapply the Kovacic algorithm.33 We describe it in the Appendix with some improvements com-pared to Refs. 34–36. Our description is self-contained and it can be useful in the study of othernonintegrability problems.

III. BIANCHI CLASS A HAMILTONIAN SYSTEMS

Homogeneous cosmological models are given by a four dimensional manifoldM 4 with apseudo-Riemannian metricds2 satisfying Einstein equations. Homogeneity of a model means thatmetric ds2 is invariant with respect to a certain three dimensional Lie groupG acting onM 4. Itis assumed that orbits of this action are three dimensional. The most important are those modelswhere the orbits ofG action are spacelike, i.e., the metric restricted to an orbit is negative define.In such caseM 4 has the structure of Cartesian product of the formM 45R3M 3 whereM 3 isan orbit of action ofG. Vector fieldsXi , i 51,2,3, which generate the action of groupG, form athree dimensional Lie algebra. A simple classification of real three dimensional Lie algebras wasdone by Bianchi in 1897. It depends on a scalaraPR and a vector (n1 ,n2 ,n3), whereniP$11,21,0% for i 51,2,3 in the following way:

@X1 ,X2#5n3X3 , @X2 ,X3#5n1X12aX2 , @X3 ,X1#5n2X21aX1 .

Table I presents Bianchi classification. This classification, as well as the corresponding cosmo-logical models, are naturally divided into two classes: to class A belong those models for whicha50, and to class B those for whichaÞ0. For a homogeneous model Einstein equations reduceto a certain dynamical system. In the case of a model belonging to class A this dynamical systemis Hamiltonian. The Hamiltonian function for this system can be written in the following form:

1730 J. Math. Phys., Vol. 42, No. 4, April 2001 Maciejewski, Strelcyn, and Szydłowski


128.192.114.19 On: Thu, 18 Dec 2014 04:53:33

H5gi j pipj11

2 (1< i , j <3

niqinjqj21

4 (i 51

3

~niqi !2, ~III.1!

where

@gi j #5F 2q12 q1q2 q1q3

q2q1 2q22 q2q3

q3q1 q3q2 2q32G .

From the physical interpretation it follows that variablesqi are non-negative. Thus our Hamil-tonian system is defined onT* Q, whereQ5R1

3 , whereR1 is the positive real axis.

IV. PROPERTIES OF BIANCHI VIII HAMILTONIAN SYSTEM

From now on we will consider only the case of the Bianchi VIII model when (n1 ,n2 ,n3)5(1,1,21). For short we called it a B~VIII ! model.

For our investigation it is important to find a particular solution for the system. To this end weselect an invariant set.

Lemma 1: For theB~VIII ! model a four dimensional manifold,

T5$~q1 ,q2 ,q3 ,p1 ,p2 ,p3!PT* Quq15q2 , p15p2%,

is invariant.Proof: We make a symplectic change of variables

@q1 ,q2 ,q3 ,p1 ,p2 ,p3#T5A@Q1 ,Q2 ,Q3 ,P1 ,P2 ,P3#T,

where

A51

2 32 2 0 0 0 0

22 2 0 0 0 0

0 0 2 0 0 0

0 0 0 1 1 0

0 0 0 21 1 0

0 0 0 0 0 2

4 . ~IV.1!

In the new variables the Hamiltonian of the B~VIII ! model has the form

H52P12Q2

22Q12~11P2

2!22Q1P1~Q2P22Q3P3!1Q2Q3~2P2P321!2 14 Q3

2~114P32!,

and thus Hamilton’s equations read

TABLE I. Bianchi classification of three-dimensional Lie algebras.

Type a n1 n2 n3 Type a n1 n2 n3

I 0 0 0 0 III 1 1 21 0II 0 1 0 0 IV 1 1 0 0

VI0 0 1 21 0 V 1 0 0 0VII 0 0 1 1 0 VIa aÞ1 1 21 0VIII 0 1 1 21 VIIa aÞ0 1 1 0IX 0 1 1 1



128.192.114.19 On: Thu, 18 Dec 2014 04:53:33

Q1522@P1Q221Q1~Q2P22Q3P3!#,

Q2522@Q2~Q1P12Q3P3!1Q12P2#,

Q352Q3@Q1P11Q2P22Q3P3#,~IV.2!

P152@Q1~11P22!1P1~Q2P22Q3P3!#,

P25Q3~122P2P3!12P1~Q2P11Q1P2!,

P35Q21 12 Q322P3~Q1P11Q2P22Q3P3!.

Manifold T is defined in the new variables as

T5$~Q1 ,Q2 ,Q3 ,P1 ,P2 ,P3!PT* QuQ150, P150%.

Direct inspection of the first and fourth equations in~IV.2! shows that indeedT is invariant. h

T is called Taub manifold and solutions lying on it are called Taub solutions. It is importantto notice that the B~VIII ! system restricted toT is integrable.

Lemma 2: TheB~VIII ! model restricted to manifoldT is a completely integrable Hamiltoniansystem.

Proof: The B~VIII ! model restricted to manifoldT is given by the following system ofdifferential equations:

Q252Q2~Q3P3!,

Q352Q3~Q2P22Q3P3!,~IV.3!

P25Q3~122P2P3!,

P35Q21 12 Q322P3~Q2P22Q3P3!.

Direct inspection shows that these equations are Hamiltonian with respect to the standard sym-plectic structure ofR4, and that the Hamiltonian reads

HTªHuT5Q2Q3~2P2P321!2 14 Q3

2~114P32!.

It is easy to verify that the function

FªQ2Q32~Q2P2!2

is a first integral of system~IV.3!. As dHT∧dFÓ0, first integral HT and F are functionallyindependent. Thus system~IV.3! is completely integrable. h

There does not exist a direct ‘‘recipe’’ which allows us to obtain an explicit form of solutionsof an integrable system. In fact, in general, it is a difficult task. Below, we demonstrate how toobtain an explicit form of a solution of system~IV.3!. To this end, we first introduce noncanonicalvariables defined as follows:

ziªqi

qi, i 51,2,3.

In variables (q1 ,q2 ,q3 ,z1 ,z2 ,z3) Bianchi class A models have the following form



128.192.114.19 On: Thu, 18 Dec 2014 04:53:33

qi5qizi , zi5~njqj2nkqk!22ni

2qi2 ,

where (i , j ,k) runs all even permutations of$1,2,3%. In new variables Hamiltonian~III.1! reads

4H5 (1< i , j <3

~zizj12ninjqiqj !2(i 51

3

~niqi !2.

Next we perform only for the B~VIII ! model. We make a transformation

xª~q1 ,q2 ,q3 ,z1 ,z2 ,z3!→Xª~Q1 ,Q2 ,Q3 ,Z1 ,Z2 ,Z3!,

given byx5AX, where matrixA is defined by~IV.1!. The explicit form of equations of motionis the following:

Q15 12 ~Q1Z21Q2Z1!,

Q25 12 ~Q1Z11Q2Z2!,

Q35Q3Z3 ,

Z1524Q1~2Q21Q3!,

Z252Q3~2Q21Q3!,

Z354Q122Q3

2 ,

and the first integral H reads

4H524Q122Q3~4Q21Q3!1Z2Z31 1

4 ~Z222Z1

2!.

Now, invariant setT is defined as follows:

T5$~Q1 ,Q2 ,Q3 ,Z1 ,Z2 ,Z3!uQ150, Z150%,

and the B~VIII ! system restricted toT has the form

Q25 12 Q2Z2 , ~IV.4!

Q35Q3Z3 , ~IV.5!

Z252Q3~2Q21Q3!, ~IV.6!

Z352Q32 . ~IV.7!

This system possesses two functionally independent first integrals

HT 52Q3~4Q21Q3!1Z2Z31 14 Z2

2 , G5Q321Z3

2 .

Equations~IV.5! and~IV.7! form a closed subset which can be easily integrated explicitly if onetakes into account thatQ3

21Z32 is its first integral. The solution has the form

Q3~ t !52AkeAt

11k2e2At , Z3~ t !5A12k2e2At

11k2e2At , ~IV.8!



128.192.114.19 On: Thu, 18 Dec 2014 04:53:33

where A>0 @because, as we already stated, the original phase space of the B~VIII ! model isT* R1

3 #, andkPR are constants of integration. Thus, the main difficulty lies in explicit integrationsof the subsystem formed by Eqs.~IV.4! and~IV.6!. However, let us notice that, using first integralHT , we can rewrite Eq.~IV.6! in the form

Z25 14 Z2

21Z3~ t !Z21Q3~ t !22B, ~IV.9!

whereB is the value of integral HT . This is the Ricatti equation, and now the main difficulty ofsolving it lies in finding its particular solution. Equation~IV.9! depends on three parameters andthus we have to find a particular solution which is valid for all values of the parameters. It mustbe noted that there is no universal method for finding a particular solution of a Ricatti equation.However, in our case we can try to find it as a function ofZ3(t). Direct calculations show that

U~ t !522Z3~ t !12AA21B

is a desired particular solution. In the above formula we assumed thatA21B>0 because we lookfor real solutions, and we take the positive value of the square root. Now, using the standardprocedure~see, e.g., Ref. 37!, we obtain a general solution of~IV.9! in the form

Z2~ t !5Z~ t !211U~ t !, ~IV.10!

where

Z~ t !5C exp@2tAA21B#21

4AA21B,

and whereC is an integration constant. Finally, integrating Eq.~IV.4! we obtain

Q2~ t !5D~11k2 exp 2tA!exp@ t~AA21B2A!#

~4CAA21B2exp@ tAA21B# !2, ~IV.11!

whereD is an integration constant.Integrating system~IV.4!–~IV.7! we introduced five constants (A,B,C,D,k). However, only

four of them can be independent. In fact, Eqs.~IV.8!, ~IV.10! and ~IV.11! give a solution ofsystem~IV.4!–~IV.7! if and only if the following condition is fulfilled:

2C~A21B!3/25kAD. ~IV.12!

For further investigation we select only nonequilibrium solutions. As it is easy to verify Eqs.~IV.8!, ~IV.10! and ~IV.11!, define an equilibrium if and only if

A5B50, or A5D5C50, or k5D5A21B50.

In our further investigations we plan to study those solutions which lie on the zero energy level.More precisely, we are interested in all real nonequilibrium solutions which lie on the invariant setH21(0)ùT. For these solutions we haveB50. Summarizing, all real nonequilibria solutions ofsystem~IV.4!–~IV.7! which lie on the Taub manifold are given by~IV.8!, ~IV.10! and ~IV.11!with (A,B,C,D,k)PP, where

Pª$~A,B,C,D,k!PR13R4 uA.0, B50, 2CA25kD%. ~IV.13!



128.192.114.19 On: Thu, 18 Dec 2014 04:53:33

V. NORMAL VARIATIONAL EQUATIONS

We can extend the B~VIII ! Hamiltonian system to a complex one, i.e., as the phase space wecan takeC6. Moreover, we can take time as a complex variable. Solutions found in the previoussection naturally extend to the complex time.

Let us take a solution (Q2(t),Q3(t),Z2(t),Z3(t)) which lies on invariant manifoldT, and letus write variational equations associated with this solution. They have the following form:

d

dth5L~ t !h, hPC6,

where

L~ t !51

2 3Z2~ t ! 0 0 Q2~ t ! 0 0

0 Z2~ t ! 0 0 Q2~ t ! 0

0 0 2Z3~ t ! 0 0 2Q3~ t !

Q~ t ! 0 0 0 0 0

0 8Q3~ t ! 8Q~ t ! 0 0 0

0 0 24Q3~ t ! 0 0 0

4 ,

and Q(t)528(2Q2(t)1Q3(t)), Q(t)5Q2(t)1Q3(t). From the definition of this manifold itfollows that variations normal toT correspond to (h1 ,h4), and thus the normal variational equa-tions are the following:

d

dth15

1

2Z2~ t !h11

1

2Q2~ t !h4 ,

d

dth4524@2Q2~ t !1Q3~ t !#h1 .

We transform this system to one equation of second order. To this end we determineh4 from thefirst equation and we substitute it in the second. After simple calculations we obtain

j1a~ t !j1b~ t !j50, ~V.1!

where

a~ t !52Z2~ t !, b~ t !54Q2~ t !22Q3~ t !21 14 Z2~ t !2,

andjªh1 .

VI. MAIN RESULT

Generally, for a Taub solution given by~IV.8!, ~IV.10! and ~IV.11! we do not know how todecide whether the identity component of the differential Galois group is Abelian or not. Thereason is connected with the fact that we do not know how to transform Eq.~V.1! to an equationwith rational coefficients. However, when such solution lies on the zero energy level H21~0! @it isequivalent to taking (A,B,C,D,k)PP in ~IV.8!, ~IV.10! and ~IV.11!# then it is rational withrespect to variablez5exp@At#. This suggests making transformationt→z in ~V.1!. In order tominimize the number of parameters we distinguish two caseskÞ0 andk50, and we define

z~ t !ªH k exp@At# for kÞ0,

exp@At# for k50.



128.192.114.19 On: Thu, 18 Dec 2014 04:53:33

Let us note that the above transformation~which is a covering of the Riemannian surface! does notchange the identity component of the differential Galois group of NVE. Now, after transformationt→z Eq. ~V.1! has the form

j91p~z!j81q~z!j50,

where

p~z!5A1a~ t~z!!

Az, q~z!5

b~ t~z!!

A2z2 .

Finally, putting

wªj exp1

2 E p~t! dt,

we transform NVE to the reduced form

w92r ~z!w50, ~VI.1!

where

r ~z!51

2

dp~z!

dz1

1

4p~z!22q~z!.

For k50 ~in this case necessarilyC50! we have

r ~z!52c2

z6 21

4z2 , where c52D

A. ~VI.2!

WhenkÞ0 then

r ~z!5s~z!

z2~z2c!4 , c54kAC, ~VI.3!

where

s~z!52 14 ~114c2!z42cz31 1

2 c2z22c3z2c2~11 14 c2!.

We prove now that NVEs withr (z) given by ~VI.2! and ~VI.3! have no Liouvillian solution.Lemma 3: For cÞ0 Eq. ~VI.1! with r(z) given by~VI.2! has no Liouvillian solution.Proof: Let us use the Kovacic algorithm presented in the Appendix. We haveG85$0%, G

5$0, %, ord(0)56, ord( )52, thusm156 andg25g51. Conditions 3, 4, 5 in the first step inthe Appendix giveL5$1%. It follows that we haven51 and we proceed to the second step. Point

3 in this step givesE`5$ 12%, and in point 5 we easily determine thata05 ic, b050, and thus

E05$ 32%. There is only one element in Cartesian productE03E` , namelye5(e0 ,e`)5( 3

2,12), and

for it we haved(e)521¹N0 . Thus the equation considered has no Liouvillian solution.hLemma 4: For cÞ0 Eq. ~VI.1! with r(z) given by~VI.3! has no Liouvillian solution.Proof: As in the previous lemma we apply the Kovacic algorithm. We haveG85$0,c%, G

5$0,c,`%, ord(0)52, ord(c)54, ord( )52, thusm154 andG25$0, %, g25g52. Conditions3–5 in the first step giveL5$1,2%. The partial fraction expansion forr (z) has the form

r ~z!5241c2

4c2z2 22~21c2!

c3z2

~11c2!2

~z2c!4 12~12c4!

c~z2c!3 221~11c2!2

c2~z2c!2 12~21c2!

c3~z2c!,



128.192.114.19 On: Thu, 18 Dec 2014 04:53:33

thus

a05241c2

4c2 , D052i

c.

The Laurent series expansion ofr (z) at infinity has the form

r ~z!52114c2

4z2 1OS 1

z3D ,

and thus

a`52 14 2c2, D`52ic.

Now, we putn51 and we go to the second step. We have

E05H 1

26

i

cJ , E`5H 1

26 icJ .

For polez5c we can find that

@Ar #c56 i~11c2!

~x2c!2 , thus ac52 i~11c2!,

r 2@Ar #c25

2~12c4!

c~z2c!3 1OS 1

~z2c!2D , thus bc5212c4

c.

From these calculations we find

Ec5H 16 i12c2

c J .

As cPR\$0%, for ePE03Ec3E` we have

Re(pPG

ep52 and Red~e!521;

thus, it never happens thatd(e)PN0 . It follows that we have to start from the second step of thealgorithm withn52. Now, we have

E05$2~12D0!12 j D0 u j 50,1,2%ùZ52,

E`5$2~12D`!12 j D` u j 50,1,2%ùZ52,

andEc54. Thus for uniquee5(2,4,2) we haved(e)522¹N0 , and we can conclude that theequation has no Liouvillian solution. h

Using the above two lemmas we prove the following.Theorem 2: The Hamiltonian system describing theB(VIII) model is not completely inte-

grable in a neighborhood of a nonequilibrium Taub solution lying on the levelH21(0).Proof: In fact, let us take a solution given by~IV.8!, ~IV.10! and ~IV.11! with

(A,B,C,D,k)PP. If k50, then, by Lemma 3, NVE has no Liouvillian solution. Thus the identitycomponent of the differential Galois group of NVE is SL(2,C), and from Theorem 1 it followsthat the B~VIII ! model is not completely integrable in a neighborhood of the selected solution. ForkÞ0 we repeat the same arguments using Lemma 4. This ends the proof. h



128.192.114.19 On: Thu, 18 Dec 2014 04:53:33

VII. REMARKS AND COMMENTS

We proved that the B~VIII ! model is not completely integrable in a neighborhood of anarbitrary nonequilibrium Taub solution lying on the zero level of the Hamiltonian. The methodapplied gives the nonintegrability result for all such solutions~note that they form a familyparametrized by three parameters!. As the theory used formulates only the necessary conditionsfor nonintegrability, one can expect that for specific values of parameters these conditions are notsatisfied. For the system studied above it is not the case. It will be interesting and important to findexamples where such phenomenon occurs. In fact, the applied theory gives only very limitedinsight into dynamical reasons of the nonintegrability, see Ref. 12. Thus, having examples forwhich this theory does not yield the nonintegrability for some parameters values we can numeri-cally investigate a neighborhood of the particular solution. It seems that the obtained local phaseportrait should show which dynamical phenomena are not ‘‘sensed’’ by the applied theory.

Let us stress that to relate the algebraic features of the system to geometry of its orbits is themain open problem in this domain.

ACKNOWLEDGMENTS

We thank J.-A. Weil~Universitede Limoges, France! for his pertinent comments concerningour presentation of the Kovacic algorithm. We thank J. J. Morales Ruiz~Universitat Polite´cnica deCatalunya! for his helpful remarks. We thank also Zbroja for her linguistic help and patience. Forthe first two authors this work was supported by program Jumelage.

The second author thanks N. Belili~Universited’Evreux, France! and D. Gutkin~UniversiteParis 13, France! for some Maple computations.

APPENDIX: KOVACIC ALGORITHM

The Kovacic algorithm gives an answer to the question if a linear second order differentialequation with rational coefficients

w91pw81qw50, p, qPC~z!, 8[d

dz, ~A1!

possesses a Liouvillian solution. For definitions, details and proofs related to differential algebra,Refs. 29, 33, and 36. It is important that an answer to this question is connected with properties ofthe identity component of the differential Galois group of~A1!.

It is known that when Eq.~A1! possesses a nonzero Liouvillian solution then all its solutionsare Liouvillian. Making a change of variables

y5w exp1

2 E p,

we transform~A1! to the reduced form

y95ry , ~A2!

where

r 5 12 p81 1

4 p22q.

The new equation has the same identity component of differential Galois group as~A1!. Thelogarithmic derivativevªy8/y of a solutiony of Equation~A2! satisfies the Riccati equation

v81v25r , ~A3!



128.192.114.19 On: Thu, 18 Dec 2014 04:53:33

and, according to Lie–Kolchin’s theorem, Eq.~A2! has a Liouvillian solution if and only if thecorresponding Riccati equation~A3! has an algebraic solution. Moreover, the degreen of theminimal polynomial for this algebraic solution belongs to

Lmaxª$1,2,4,6,12%.

The differential Galois groupG of ~A2! is an algebraic subgroup of SL(2,C) and its identitycomponentG0 is of one of the following forms

Case 1:G0 is triangularizable; for this case Eq.~A2! is reducible and has a solution of theform y5exp*v, wherevPC(z), i.e., Riccati equation~A3! has a rational solution (n51).

Case 2:G0 is imprimitive; for this case Eq.~A2! has a solution of the formy5exp*v, wherev is algebraic overC(z) of degree 2, i.e., Riccati equation~A3! has an algebraic solution of degreen52.

Case 3:G0 is primitive and finite; for this case all solutions of Eq.~A2! are algebraic andRiccati equation~A3! has an algebraic solution of degreenP$4,6,12%.

Case 4:G05SL(2,C) and Eq.~A2! has no Liouvillian solution, i.e. Riccati equation~A3! hasno algebraic solution.

Now, we present the algorithm. First, we fix notation. The set of non-negative integers isdenoted byN0 . We define onLmax function h in the following way:

h~1!51, h~2!52, h~4!53, h~6!52, h~12!51.

We write r PC(z) in the form

r 5s

t, s, tPC@z#,

wheres and t are relatively prime andt is monic. The algorithm consists of three steps.First step~1! Let G8ª$cPC u t(c)50%, GªG8ø$`%. The order ord(c) of cPG8 is equal to the mul-

tiplicity of c as a root oft; the order of infinity is defined by

ord~`!ªmax~0,41degs2degt !.

~2! Define

m1ªmax

cPG

ord~c!.

For i PN0 let

G iª$cPG u ord~c!5 i %,

G i8ªG i\$`%, g iªcardG i , and gªg21 (oddk

3<k<m1

gk .

~3! DefineL8,Lmax by the following rules:

1PL8⇔g5g2 ,

2PL8⇔g>2,

4,6,12PL8⇔m1<2.

~4! For eachcPG1øG2 find the Laurent series expansion ofr aroundc in the form



128.192.114.19 On: Thu, 18 Dec 2014 04:53:33

r 5ac

~z2c!2 1bc

z2c1O~1!,

for cPG18øG28 , and

r 5a`

z2 1b`

z3 1OS 1

z4D ,

for c5`. PutDcªA114ac.~5! If m1.2, then L5L8. If m1<2 and ;cPG1øG2 DcPQ, then L5L8 else L

5L8\$4,6,12%.~6! If L5B, then go toEnd else assignn to the smallest element ofL.Second step~1! If `PG0 , thenE`ª$h(n)k u k50, . . . ,n%.~2! For eachcPG1 , define setEcª$nh(n)%.~3! Whenn51, for eachcPG2 , define the set

EcªH 1

2~16Dc!J .

~4! Whenn>2, for eachcPG2 , define the set

EcªH h~n!

2~n2~n22 j !Dc! U j 50, . . . ,nJ ùZ.

If at least one setEc is empty then go toContinue.~5! Whenn51, for eachcPG2k , with k>2, compute rational function@Ar #c defined up to

sign by the following conditions:

• for cPG2k8

@Ar #c5ac

~z2c!k 1 (2< j <k21

sj ,c

~z2c! j ,

r 2@Ar #c25

bc

~z2c!k11 1OS 1

~z2c!kD ,

• for c5`

@Ar #`5a`zk221 (0< j <k23

sj ,`zj ,

r 2@Ar #`2 52b`zk231O~zk24!.

Now define the setEc by

EcªH 1

2 S k1«bc

acD U «561J ,

and function

sign:Ec→$11,21%,



128.192.114.19 On: Thu, 18 Dec 2014 04:53:33

signS 1

2 S k1«bc

acD DªH « if bcÞ0,

11 if bc50.

~6! Whenn52 for eachcPGk , with k>3, define the setEc by Ecª$k%.Third step~1! For eache5$ec%cPG in Cartesian productEª)cPGEc compute

d~e!5n21

h~n! (cPG

ec .

~2! Select elementse for which

~a! d(e)PN0 , and~b! whenn52 or n56, thene has an even number of components which are odd integers, and~c! whenn54, thene has at least two components which are not divisible by 3 and the sum of

all components which are not divisible by 3 is divisible by 3.

If no such element exists, then go toContinue.~3! For each selected elemente put

u51

h~n! (cPG8

ec

z2c1dn

1 (cPG2kk.1

sign~ec!@Ar #c ,

wheredn1 is the Kronecker symbol.

~4! For each (e,u) decide if there exists a monic polynomialP of degreed5d(e) satisfyingthe following system of equations,

Pn52P,

Pi 2152Pi82uPi2~ i 11!~n2 i !rPi 11 , for n> i>0,

P2150,

and if so find it~in the above formulasPi8 denotes the derivative ofPi with respect toz!.Output: If a pair (u,P) is found, then equationy95ry possesses the Liouvillian solution

h5exp*v wherev is a solution of the following irreducible algebraic equation

(i 51

nPi

~n2 i !!v i50,

elseContinue: if n is not the greatest element inL then assign ton the next value inL and go to

Second stepelseEnd: equationy95ry has no Liouvillian solution.

1. Comments

A formulation of the Kovacic algorithm given in Ref. 38 is alternative to its original form andto that presented previously. It seems that it is much more convenient for computer implementa-tion. However, for differential equations with simple structure of singularities, and depending onparameters it seems that the previous form of the algorithm is well suited. Let us note also that, asit was explained in Ref. 38, the polynomialP which appears in the point 4 of the third stepsatisfies a linear differential equation of ordern11 ~this equation is isomorphic, as a differentialoperator, of thenth symmetric power of the investigated equation!.



128.192.114.19 On: Thu, 18 Dec 2014 04:53:33

In the original formulation of the algorithm23 consisting in fact, of three separated algorithmscorresponding to cases 1, 2 and 3, each of them repeats similar steps. In Ref. 36 one can find amodification of the original formulation unifying and improving three algorithms in one. Thisform is very convenient for applications. However, there are errors in the algorithm. In fact, for

r ~z!523

16z2 22

9~z21!2 13

16z~z21!,

the equationy95r (z)y has a Liouvillian solution~see Ref. 33, Example 1, p. 23!, but algorithmin Ref. 36 finds no such solution. Our analysis showed that there are three errors. Namely, we havethe following.

~1! Conditions 5 in the first step of the algorithm~see Ref. 36, p. 215! are wrong. Theseconditions do not appear in earlier published versions of the algorithm34,35. However, when cor-rectly stated they are important—they are necessary conditions for the existence of an algebraicsolution ~case 3!. Their meaning is the following. If case 3 occurs, then polescPG8 have ordernot greater than 2 and for allcPG8 we haveDcPQ. Moreover,c5` has order not greater then2. This is equivalent to(cPG

18øG28bc50. Finally, as it was shown in Ref. 33~p. 11!,

a`5 (cPG28

ac1 (cPG18øG28

cbc .

Thus, the proper necessary condition isD`PQ.~2! In the third step condition 2~b!, n54 in Ref. 36~p. 216! is not correct. In fact, for

r ~z!525z2127

36~z221!2 ,

the equationy95r (z)y has an algebraic solution and casen54 occurs~see Ref. 33, Example 2,p. 25!. Applying the algorithm of Ref. 36 we find forn54, e5(4,4,4)PE13E213E` . For thise we haved(e)50, but, according to the mentioned condition, we have to reject it from furthercalculations. However, exactly for thise, using the original algorithm we find the desired solution.This error appears also in the earlier version of the algorithm.34,35

~3! In the third step there is an additional condition forn54 in 2~b!. Namely, at least twocomponents ofe are divisible by 3. The example in the previous point shows that it is not a correctcondition. In the earlier version of the algorithm the mentioned condition reads ‘‘... at least twocomponents ofe are not multiples of 3,’’ and this is the correct condition.Let us explain how the proper condition for components of vectorse in the third step can bededuced. Forn52, 4 and 6 we can choose the fundamental solution (j,h) of the equationy95ry in such a way thatun

h(n)PC(z), whereu2ªjh, u4ªh418hj3, andu6ªjh52j5h. More-over, from the structure of differential Galois groups for the respective cases it follows thatun¹C(z) for n52, 4 and 6. Thus writing

unh(n)5 )

cPG8~z2c!ec, ecPZ,

we deduce that forn52 andn56 at least oneec for cPG8 is an odd integer, and forn54 at leastoneec is not divisible by 3. Moreover, we have

d~e!5n21

h~n! (cPG

ec , ecPZ,

and this implies the necessary conditions in the presented algorithm.



128.192.114.19 On: Thu, 18 Dec 2014 04:53:33

Finally let us notice that in Ref. 39 necessary conditions for the existence of a Liouvilliansolution of third and second order linear differential equations were formulated. This article givesalso a clear explanation of the origin of arithmetic conditions which appear in the present formu-lation of the algorithm.

1A. J. Maciejewski and M. Szydlowski, Celest. Mech. Dyn. Astron.73, 17 ~1999!.2D. F. Chernoff and J. D. Barrow, Phys. Rev. Lett.50, 134 ~1983!.3R. Cushman and J. Sńiatycki, Rep. Math. Phys.36, 75 ~1995!.4N. J. Cornish, inRecent Developments in Theoretical and Experimental General Relativity, Gravitation, and RelativisticField Theories, edited by T. Piran and R. Ruffini~World Scientific, Singapore, 1999!, pp. 613–616.

5N. J. Cornish and J. J. Levin, inRecent Developments in Theoretical and Experimental General Relativity, Gravitation,and Relativistic Field Theories, edited by T. Piran and R. Ruffini~World Scientific, Singapore, 1999!, pp. 616–619.

6A. Ramani, B. Grammaticos, and T. Bountis, Phys. Rep.180, 159 ~1989!.7G. Contopoulos, B. Grammaticos, and A. Ramani, J. Phys. A26, 5795~1993!.8G. Contopoulos, B. Grammaticos, and A. Ramani, inDeterministic Chaos in General Relativity (Kananaskis, AB, 1993)~Plenum, New York, 1994!, pp. 423–432.

9G. Contopoulos, B. Grammaticos, and A. Ramani, J. Phys. A27, 5357~1994!.10A. Latifi, M. Musette, and R. Conte, Phys. Lett. A194, 83 ~1994!.11G. Contopoulos, B. Grammaticos, and A. Ramani, J. Phys. A28, 5313~1995!.12J. J. Morales Ruiz,Differential Galois Theory and Non-integrability of Hamiltonian Systems~Birkhauser Verlag, Basel,

1999!.13S. L. Ziglin, Funktsional. Anal. i Prilozhen.16, 30 ~1982!.14S. L. Ziglin, Funktsional. Anal. i Prilozhen.17, 8 ~1983!.15A. Braider and R. C. Churchill, J. Diff. Eqns.2, 451 ~1990!.16R. C. Churchill and D. L. Rod, SIAM~Soc. Ind. Appl. Math.! J. Math. Anal.22, 1790~1991!.17J.-P. Françoise and M. Irigoyen, C. R. Acad. Sci., Ser. I: Math.311, 165 ~1990!.18J.-P. Françoise and M. Irigoyen, J. Geom. Phys.10, 231 ~1993!.19H. Ito, Kodai Math. J.1, 120 ~1985!.20H. Ito, Z. Angew. Math. Phys.38, 459 ~1987!.21H. Ito, Bol. Soc. Brasil. Mat.21, 95 ~1990!.22D. L. Rod, Canadian Math. Soc., Conf. Proc.8, 599 ~1987!.23D. L. Rod, Contemp. Math.81, 259 ~1988!.24H. Yoshida, Physica D21, 163 ~1986!.25H. Yoshida, Physica D29, 128 ~1987!.26H. Yoshida, Celest. Mech.44, 313 ~1988!.27H. Yoshida, Phys. Lett. A141, 108 ~1989!.28F. Beukers, inFrom Number Theory to Physics~Les Houches, 1989! ~Springer, Berlin, 1992!, pp. 413–439.29I. Kaplansky,An Introduction to Differential Algebra~Hermann, Paris, 1976!, 2nd ed., actualite´s Scientifiques et Indus-

trielles, No. 1251, Publications de l’Institut de Mathe´matique de l’Universite´ de Nancago, No. V.30A. R. Magid, Lectures on Differential Galois Theory, Vol. 7 of University Lecture Series~American Mathematical

Society, Providence, RI, 1994!.31J.-P. Ramis and J. Martinet, inComputer Algebra and Differential Equations~Academic, London, 1990!, pp. 117–214.32M. F. Singer, inComputer Algebra and Differential Equations~Academic, London, 1990!, pp. 3–57.33J. J. Kovacic, J. Symbolic Comput.2, 3 ~1986!.34A. Duval, in Differential Equations and Computer Algebra, edited by M. F. Singer~Academic, London, 1991!, pp.

113–130.35A. Duval and M. Loday-Richaud,A propos de l’algotitme de Kovac˘ic, Tech. Rep., Universite´ de Paris-Sud, Mathe´ma-

tiques, Orsay, France~1989!.36A. Duval and M. Loday-Richaud, Appl. Algebra Engrg. Comm. Comput.3, 211 ~1992!.37E. Kamke,Hanbook on Ordinary Differential Equations~Nauka, Moscow, 1976! ~in Russian!.38F. Ulmer and J.-A. Weil, J. Symbolic Comput.22, 179 ~1996!.39M. F. Singer and F. Ulmer, Appl. Algebra Engrg. Comm. Comput.6, 1 ~1995!.



128.192.114.19 On: Thu, 18 Dec 2014 04:53:33

nonintegrability of bianchi viii hamiltonian system

Documents