Singularities of bihamiltonian systems and the multidimensional rigid body

Singularities of bihamiltoniansystems and themultidimensional rigid bodybyAnton IzosimovA Doctoral ThesisSubmitted in partial fulfilment of the requirements for theaward of Doctor of Philosophy of Loughborough University24th January 2012c A Izosimov 2012Abstract.Two Poisson brackets are called compatible if any linear combination of thesebrackets is a Poisson bracket again. The set of non-zero linear combinationsof two compatible Poisson brackets is called a Poisson pencil.

A system iscalled bihamiltonian (with respect to a given pencil) if it is hamiltonian withrespect to any bracket of the pencil. The property of being bihamiltonian isclosely related to integrability. On the one hand, many integrable systemsknown from physics and geometry possess a bihamiltonian structure. On theother hand, if we have a bihamiltonian system, then the Casimir functions ofthe brackets of the pencil are commuting integrals of the system. We considerthe situation when these integrals are enough for complete integrability. As itwas shown by Bolsinov and Oshemkov, many properties of the system in thiscase can be deduced from the properties of the Poisson pencil itself, withoutexplicit analysis of the integrals. Developing these ideas, we introduce anotion of linearization of a Poisson pencil.

In terms of linearization, we givea criterion for non-degeneracy of a singular point and describe its type.These results are applied to solve the stability problem for a free multidimensional rigid body.Keywords.Bihamiltonian systems, integrable systems, singularities, multidimensionalrigid bodyiiAcknowledgements.I would like to express many thanks to my supervisor Professor Alexey Bolsinov for pointing out the problem, fruitful discussions and useful comments.Thanks to all the staff of the School of Mathematics at Loughborough University for the discussion-friendly and encouraging environment, constantseminar and conference activity, and useful and effective administrative support.I also would like to thank the Overseas Research Students Awards Schemeand the School of Mathematics of Loughborough University for the financialsupport during my PhD studies.I am also grateful to the government of the Russian Federation for thelimited financial support (under the agreement 11.G34.31.0054).iiiContents1 Introduction11.1Statement of the problem .

. . . . . . . . . . . . . . . . . . .11.2Structure of the thesis . . . . . . . . . . . . . . . . . . . . . .51.3Integrability and non-degeneracy . . . . . . . . . . . . . . . .71.4Jordan-Kronecker theorem . . . . . . . . . . . . . .

. . . . . . 101.5Bihamiltonian systems and integrals in involution . . . . . . . 121.6Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Definitions and the non-degeneracy criteria152.1Linear pencils . . . . . . . . . . . . .

. . . . . . . . . . . . . . 152.2Singularities associated with integrable linear pencils . . . . . 182.3Linearization of a Poisson pencil . . . . . . . . . . . . . . . . . 242.4The non-degeneracy criterion . . . . . . . . . . . . . . . . . . 263 Zero order theory303.1Geometry of a pair of skew-symmetric forms on a vector space 303.2The recursion operator .

. . . . . . . . . . . . . . . . . . . . . 364 First order theory394.1Definition of the operator Df P4.2Operators Df Pα for f ∈ F . . . . . . . . . . . . . . . . . . . . 42iv. . . . . . . . . . . . . . . . . 394.3Operator Df P and linearizations of hamiltonian vector fields . 465 Proof of the main theorems495.1Non-degeneracy of linear pencils . . .

. . . . . . . . . . . . . . 495.2Classification of non-degenerate linear pencils: the complex case 515.3Classification of non-degenerate linear pencils: the real case . . 565.4Proof of the second part of Theorem 6 . . . . . . . . . . . . . 595.5Proof of the non-degeneracy criterion for arbitrary pencils . . 606 Multidimensional rigid body646.1Introduction . . .

. . . . . . . . . . . . . . . . . . . . . . . . . 646.2The equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.3Description of relative equilibria . . . . . . . . . . . . . . . . . 706.4Parabolic diagram of a regular relative equilibrium6.5Stability theorems .

. . . . . . . . . . . . . . . . . . . . . . . . 756.6The bihamiltonian structure . . . . . . . . . . . . . . . . . . . 826.7Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.8The bad set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866.9Complete integrability . . . . . . .

. . . . . . . . . . . . . . . 87. . . . . . 736.10 Rank zero singular points . . . . . . . . . . . . . . . . . . . . 886.11 Non-degeneracy and type theorems . . . . . . . . . . . . . . . 916.12 Non-degeneracy: scheme of the proof . . . . . . . . . . . . . . 926.13 Description of Λ(M ) .

. . . . . . . . . . . . . . . . . . . . . . 926.14 When is the pencil diagonalizable? . . . . . . . . . . . . . . . 966.15 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . 976.16 Proof of non-degeneracy and type theorems . . . . . .

. . . . 1036.17 Non-resonancy. . . . . . . . . . . . . . . . . . . . . . . . . . 1056.18 Proof of stability theorems . . . . . . . . . . . . . . . . . . . . 110vChapter 1Introduction1.1Statement of the problemA system is called bihamiltonian if it is hamiltonian with respect to a wholeone-dimensional family of Poisson brackets (“Poisson pencil”). It is wellknown that bihamiltonian systems are often completely integrable. The relation between bihamiltonian structure and integrability was first noted byF.Magri in his paper [25], where the infinite-dimensional situation was studied. The observation made by Magri is the following: if a (partial differential)equation admits two Hamiltonian representations, then we can construct aninfinite sequence of commuting integrals.The finite-dimensional situation was probably first considered by Gelfandand Dorfman in [17].Theorem 1 (Gelfand-Dorfman, [17]).

Let A, B be two non-degenerate compatible Poisson structures and v be a vector field hamiltonian with respect toboth of them, i.e.v = Adf1 = Bdf0for some functions f0 , f1 . Then, provided that the first cohomology group of1the manifold vanishes, it is possible to find functions f2 , f3 , . . . such thatAdfi+1 = Bdfiand all fi are integrals of v, which are in involution with respect to both Aand B.Therefore, if a vector field is bihamiltonian with respect to two compatiblenon-degenerate brackets, we can find many integrals in involution.

Usually,these integrals are enough for complete integrability.Another idea for constructing integrals of a bihamiltonian system isbrought by the followingTheorem 2 (Magri-Morosi, [26]). Let v be a vector field hamiltonian withrespect to two non-degenerate Poisson structures A and B. Let R = A−1 B.Then• Traces of powers of R are integrals of v. They are in involution withrespect to both A and B.• If the multiplicities of all the eigenvalues of R are equal to two, then vis Liouville integrable.The situation we are going to discuss is different: let the brackets forming a pencil be degenerate. In this case another idea can be applied: if v isbihamiltonian with respect to a pencil, then all Casimir functions of all maximal rank brackets of the pencil are integrals of v in involution. To author’sknowledge, this was first noted by Reiman and Semenov-Tyan-Shanskii in[36].

However, this construction can be also considered as a generalisation ofthe argument shift method, introduced by A.S.Mischenko and A.T.Fomenkoin [30], and the Manakov construction of the integrals in the Euler case ofmultidimensional rigid body dynamics (see [27] and Chapter 6 of this thesis).2It often happens that the integrals obtained from the Casimir functions ofthe brackets of the pencil are enough for complete integrability of the system.The necessary and sufficient condition for this was obtained by A.V.Bolsinovin [5].On the other hand, many integrable systems appearing in geometry, mechanics and physics possess a bihamiltonian structure (see, for example,[6, 13]).

In spite of the fact that the complete integrability of these systemswas proved before the bihamiltonian structure was found, the bihamiltonianrepresentation is useful for the qualitative analysis of the dynamics, as it wasshown by A.V.Bolsinov and A.A.Oshemkov in [8].The theory of qualitative (or topological) analysis of integrable hamiltonian systems is due to the works of A.T.Fomenko and his school (see [15,16, 7]), as well as L.M.Lerman and Ya.L.Umanskiy (see [20, 21, 22, 23, 24]),and M.P.Kharlamov (see [18]).

The idea of this theory goes back to theArnold-Liouville theorem (see [2, 7]), which says that the phase space of anintegrable hamiltonian system is foliated almost everywhere into invarianttori, and the dynamics on these tori is quasi-periodic. Consequently, if oneaims to understand the dynamics of an integrable hamiltonian system, itis very important to study the topology of the foliation into tori (so-calledLiouville foliation). The second idea, successfully applied by the mentionedauthors, can be formulated as follows: the topology of a Liouville foliation ismainly defined by the singularities of the system.Therefore, the theory ofqualitative analysis of integrable hamiltonian systems is mainly the theoryof singularities of such systems.Singular points of an integrable systems are those points in which theintegrals become dependent. According to the general scheme of the theoryof integrable hamiltonian systems, the first thing we should do to describe the3behavior of the system is to find the singular points.

In order to do this, oneshould calculate the Jacobi matrix for the set of integrals and find the pointsat which the rank of this matrix drops, which can be very complicated in largedimensions. However, if the system possesses a bihamiltonian structure, thisreduces to a procedure of describing the singular points of the pencil, i.e. theunion of singular points of all brackets of the pencil, which is much easier inexamples than the calculations with the Jacobi matrix (A.V.Bolsinov, [5]).As it was shown by A.V.Bolsinov and A.A.Oshemkov in [8], bihamiltonianstructure also allows us to simplify the analysis of topology of Liouville foliation in the neighbourhood of a singular point.