Singularities of bihamiltonian systems and the multidimensional rigid body, страница 2

In particular, bihamiltonianstructure allows us to prove non-degeneracy of singular points.Non-degenerate singular points are, in some sense, generic singular points.The notion of non-degenerate singular point of an integrable system is analogous to the notion of Morse singular point of a smooth function. Instead ofthe Morse lemma we have the Eliasson theorem here: the Liouville foliationin the neighbourhood of a singular point is symplectomorphic to the foliationgiven by the quadratic parts of the integrals.

The complete invariant of theLiouville foliation in the neighbourhood of a non-degenerate singular pointis the (Williamson) type of the point - three non-negative integers ke , kh , kfsuch thatke + kh + 2kf + r = n,where r is the rank of a point and n is the number of degrees of freedom.The notion of type of a singular point of an integrable system is analogousto the notion of an index of a Morse singular point1 .Thus knowing the type of a singular point makes it possible to describe1See Section 1.3 for precise definitions of non-degeneracy and type.4the Liouville foliation as well as dynamics in the neighbourhood of a point2 .Therefore, the first thing one should do after describing the set of singularpoints is to check whether these points are non-degenerate and find theirtype. For an arbitrary system this involves some non-trivial calculations.To solve the problem for bihamiltonian systems, we introduce the notion oflinearization of a Poisson pencil at a singular point, which is again a Poissonpencil, but a linear one.

The problem of non-degeneracy and type for aninitial pencil is reduced to the same problem for a linearized one, while thelinear problem can be easily solved in algebraic terms.We emphasise that while integrable systems are considered to be themost “symmetric” among all dynamical systems, the systems possessing abihamiltonian structure are even more “symmetric”. Therefore, applyinggeneral methods of the theory of integrable systems to bihamiltonian systemsseems to be unreasonable, and a separate theory should be developed. Anattempt to develop such a theory is made in this thesis.1.2Structure of the thesisIn Section 1.3 we give the definition of a complete family of integrals and anintegrable hamiltonian system.

Further we give definitions of non-degeneracyand type and formulate the Eliasson theorem about the linearization of aLiouville foliation in a neighbourhood of a singular point.2For example, knowing the type of a point, we can study its Lyapunov stability. Supposewe have a non-degenerate singular point, which is a fixed point of our system. Then,provided the system is non-resonant, this point is stable if and only if it has rank zeroand it has a so-called elliptic type, which means that kh = kf = 0. The same is true forperiodic trajectories, with the only difference that rank zero should be replaced with rankone.5In Section 1.4 we formulate the Jordan-Kronecker theorem about thenormal form of two skew-symmetric bilinear forms. This theorem plays avery important role in the theory of bihamiltonian systems.In Section 1.5 we discuss the notion of a bihamiltonian system and givethe construction of the involutive family of integrals associated with a bihamiltonian structure.In Section 1.6 we give A.Bolsinov completeness criterion for the systemconstructed in Section 1.5 and formulate the question about non-degeneracyand type.In Sections 2.1-2.3 we introduce the notion of a linear pencil and linearization of an arbitrary pencil.

The non-degeneracy and type problem isbeing solved for linear pencils.Section 2.4 is devoted to the main result of the thesis: the criterion fornon-degeneracy for a general bihamiltonian system and the type determination procedure.Chapters 3 and 4 contain technical statements and constructions, neededfor the proofs. Chapter 3 contains those which involve only zero order termsof the brackets (in other words, we consider a pencil of skew-symmetric formson a vector space), while chapter 4 deals with the first order terms.Proofs themselves are given in Chapter 5.In the last Chapter 6 we apply our results to study stability of relativeequilibria of a free multidimensional rigid body.61.3Integrability and non-degeneracyLet ẋ = sgrad H be a hamiltonian system on a symplectic manifold (M 2n , ω),wheresgrad H = ω −1 dH.Let F be a family of pairwise commuting integrals of the system.

It will beconvenient to assume that F is a vector space, i.e. is closed under additionand multiplication by numbers. If it is not so, we can always replace F withthe space linearly spanned by F.Definition 1.1. F is said to be complete on M 2n if for almost all x ∈ M 2n the spacedF(x) = {df (x), f ∈ F} is maximal isotropic with respect to thePoisson bracket ω −1 , i.e. has dimension n.2. If additionally the vector fields sgrad f, f ∈ F are complete, the systemẋ = sgrad H is called completely integrable. The foliation of M into theconnected components of the common level sets {F = const} is calledin this case the Liouville foliation.Remark 1.3.1.

This definition coincides with the classical one if it is possibleto choose n functions in F such that their differentials span dF. This cannotalways be done globally, but can be done locally, which guarantees that theLiouville theorem still holds for the systems integrable in our sense.Definition 2. A point x is called singular for a given integrable hamiltoniansystem if dim dF(x) < n (i.e.

dF(x) is not maximal isotropic). The numberrank x = dim dF(x) is called the rank of a singular point x. A fiber of Liouville foliation, which contains at least one singular point, is called singular.All other fibers are called regular.7The Liouville theorem (see [2, 7]) claims that all compact regular fibers ofa Liouville foliation are tori, and the dynamics on these tori is conditionallyperiodic.Despite the fact that almost all fibers of a Liouville foliation are regular, it is also important to understand the topology and dynamics in theneighbourhood of singular fibers due to the following reasons:1. Singular fibers correspond to singular regimes of motion.

In particular,fixed points of a system always belong to singular fibers.2. It is mainly singular fibers which define the global topology of a system.Let us now define what a non-degenerate singular point is.Suppose that f ∈ F, df (x) = 0. Then we can consider the linearizationof the vector field sgrad f at the point x. Denote it by Af . Since the flowdefined by sgrad f preserves the symplectic structure, Af ∈ sp(Tx M ).Now consider the space W = {sgrad f (x), f ∈ F}. Since the functionsin F are in involution, all operators Af vanish on W . Consequently, we canconsider Af as operators on W ⊥ /W .

Since W is isotropic, W ⊥ /W carriesa natural symplectic structure and Af ∈ sp(W ⊥ /W ). Also note that all Afcommute, therefore the setAF = {Af , f ∈ F, df (x) = 0}is a commutative subalgebra in sp(W ⊥ /W ).Definition 3. A singular point x is called non-degenerate, if the subalgebraAF constructed above is a Cartan subalgebra in sp(W ⊥ /W ).If A is an element of a Cartan subalgebra h in sp, then its eigenvalues8have the form± λ1 i, . . . , ±λke i,± ν1 , . .

. , ±νkh ,± µ1 ± ξ1 i, . . . , ±µkf ± ξkf i.The triple (ke , kh , kf ) is the same for almost all A ∈ h. Let us call this triplethe (Williamson) type of the Cartan subalgebra h. All Cartan subalgebrasof the same type are conjugate to each other (Williamson, [39]).Definition 4.

The type of a singular point x is the type of the Cartansubalgebra AF ⊂ sp(W ⊥ /W ) constructed above.It is easy to see that for every non-degenerate singular point x the following equality holds: ke + kh + 2kf = n − rank x.Let us now state the Eliasson theorem about the linearization of a Liouville foliation in a neighbourhood of a non-degenerate singular point.Definition 5.1. The foliation given by the function p2 + q 2 in a neighbourhood of theorigin in (R2 , dp ∧ dq) is called an elliptic singularity.2.

The foliation given by the function pq in a neighbourhood of the originin (R2 , dp ∧ dq) is called a hyperbolic singularity.3. The foliation given by the commuting functions p1 q1 + p2 q2 , p1 q2 − q1 p2in a neighbourhood of the origin in (R4 , dp ∧ dq) is called a focus-focussingularity.Theorem 3 (Eliasson-Miranda, see [11, 12, 28] for proof, see also [7]). ALiouville foliation in a neighbourhood of a non-degenerate singular point of9rank r and type (ke , kh , kf ) is locally fiberwise symplectomorphic to a directproduct of ke elliptic, kh hyperbolic and kf focus-focus singularities, multipliedby a trivial foliation Rr × Rr .1.4Jordan-Kronecker theoremIt is well known that two bilinear symmetric forms, one of which is positivedefinite, can be simultaneously diagonalized.