Action-angle coordinates for integrable systems on Poisson manifolds
aa r X i v : . [ m a t h . S G ] J un ACTION-ANGLE COORDINATES FOR INTEGRABLESYSTEMS ON POISSON MANIFOLDS
CAMILLE LAURENT-GENGOUX, EVA MIRANDA , AND POL VANHAECKE Abstract.
We prove the action-angle theorem in the general, and mostnatural, context of integrable systems on Poisson manifolds, thereby gen-eralizing the classical proof, which is given in the context of symplecticmanifolds. The topological part of the proof parallels the proof of thesymplectic case, but the rest of the proof is quite different, since we arenaturally led to using the calculus of polyvector fields, rather than dif-ferential forms; in particular, we use in the end a Poisson version of theclassical Carath´eodory-Jacobi-Lie theorem, which we also prove. At theend of the article, we generalize the action-angle theorem to the settingof non-commutative integrable systems on Poisson manifolds.
Contents
1. Introduction 22. The Carath´eodory-Jacobi-Lie theorem for Poisson manifolds 62.1. The theorem 62.2. A counterexample 83. Action-angle coordinates for Liouville integrable systems onPoisson manifolds 103.1. Standard Liouville tori of Liouville integrable systems 103.2. Foliation by standard Liouville tori 103.3. Standard Liouville tori and Hamiltonian actions 133.4. The existence of action-angle coordinates 184. Action-angle coordinates for non-commutative integrable systemson Poisson manifolds 204.1. Non-commutative integrable systems 20
Date : October 24, 2018.2000
Mathematics Subject Classification.
Key words and phrases.
Action-angle coordinates, Integrable systems, Poissonmanifolds. Research supported by a Juan de la Cierva contract and partially supported by theDGICYT/FEDER, project number MTM2006-04353 (Geometr´ıa Hiperb´olica y Geometr´ıaSimpl´ectica). Partially supported by a European Science Foundation grant (MISGAM), a MarieCurie grant (ENIGMA) and an ANR grant (GIMP). , AND POL VANHAECKE Introduction
The action-angle theorem is one of the basic theorems in the theory ofintegrable systems. In this paper we prove this theorem in the general, andmost natural, context of integrable systems on Poisson manifolds.We recall that a Poisson manifold ( M, Π) is a smooth manifold M onwhich there is given a bivector field Π, with the property that the bracketon C ∞ ( M ), defined for arbitrary smooth functions f and g on M by { f, g } := Π(d f, d g )is a Lie bracket, i.e., it satisfies the Jacobi identity. On a Poisson manifold( M, Π), the Hamiltonian operator, which assigns to a function on M a vectorfield on M , is defined naturally by contracting the bivector field with thefunction (the “Hamiltonian”): for h ∈ C ∞ ( M ) its Hamiltonian vector fieldis defined by X h := {· , h } = − ı d h Π . (1.1)Two important consequences of the Jacobi identity for {· , ·} are that the(generalized distribution) on M , defined by the Hamiltonian vector fields X h is integrable, and that the Hamiltonian vector fields which are associ-ated to Poisson commuting functions (usually called functions in involution)are commuting vector fields. The main examples of Poisson manifolds aresymplectic manifolds and the dual of a (finite-dimensional) Lie algebra, butthere are many other examples, which come up naturally in deformationtheory, the theory of R -brackets, Lie-Poisson groups, and so on. Poisson’soriginal bracket on C ∞ ( R r ), given for smooth functions f and g by { f, g } := r X i =1 (cid:18) ∂f∂q i ∂g∂p i − ∂g∂q i ∂f∂p i (cid:19) , (1.2)is still today of fundamental importance in classical and quantum mechanics,and in other areas of mathematical physics. Many examples of integrableHamiltonian systems are known in the context of Poisson manifolds whichare not symplectic. For instance the Kepler problem [20], Toda lattices [1]and the Gelfand-Cetlin systems [11, 10].One of the main uses of the Poisson bracket is the integration of Hamil-ton’s equations, which are the equations of motion which describe a clas-sical mechanical system on the phase space R r ≃ T ∗ R r , defined by a CTION-ANGLE COORDINATES 3
Hamiltonian h (the energy, viewed as a function on phase space); their so-lutions are the integral curves of the Hamiltonian vector field X h , definedby (1.1) with respect to the Poisson bracket (1.2). The fundamental Liou-ville theorem states that it suffices to have r independent functions in invo-lution ( f = h, f , . . . , f r ) to quite explicitly (i.e., by quadratures) integratethe equations of motion for generic initial conditions. Moreover, assumingthat the so-called invariant manifolds, which are the (generic) submanifoldstraced out by the n commuting vector fields X f i , are compact, they are (dif-feomorphic to) tori T r = R r / Λ, where Λ is a lattice in R r ; on these tori,which are known as Liouville tori, the flow of each of the vector fields X f i is linear, so that the solutions of Hamilton’s equations are quasi-periodic.The classical action-angle theorem goes one step further: under the abovetopological assumption, there exist on a neighbourhood U of every Liouvilletorus functions σ , . . . , σ r and R / Z -valued functions θ , . . . , θ r , having thefollowing properties:(1) The map Φ, defined by Φ := ( θ , . . . , θ r , σ , . . . , σ r ) is a diffeomor-phism from U onto the product T r × B r , where B r is an r -dimensionalball;(2) Φ is a canonical map: in terms of θ , . . . , θ r , σ , . . . , σ r the Poissonstructure takes the same form as in (1.2) (upon replacing q i by θ i and p i by σ i );(3) Under Φ, the Liouville tori in U correspond to the fibers of thenatural projection T r × B r → B r .The proof of this theorem goes back to Mineur [13, 14, 15]. A proof in thecase of a Liouville integrable system on a symplectic manifold was given byArnold [2]; see also [4, 7, 12]. As established in [11], action-angle coordinatesalso appear naturally in geometric quantization, for, when an integrablesystem is interpreted as a polarization, action-angle coordinates determinethe so-called Bohr-Sommerfeld leaves: the latter are in particular explicitelydescribed for the Gelfand-Cetlin system in [11].In the context of Poisson manifolds, the Liouville theorem still holds, upto two adaptations: one needs to take into account the Casimirs (functionswhose Hamiltonian vector field are zero) and the singularities of the Poissonstructure (the points where the rank of the bivector field drops); for a precisestatement and a proof, see [1, Ch. 4.3]. As we show in this paper, the action-angle theorem takes in the case of Poisson manifolds the following form Theorem 1.1.
Let ( M, Π) be a Poisson manifold of dimension n and (max-imal) rank r . Suppose that F = ( f , . . . , f s ) is an integrable system on ( M, Π) , i.e., r + s = n and the components of F are independent and ininvolution. Suppose that m ∈ M is a point such that (1) d m f ∧ . . . ∧ d m f s = 0 ; An equivalent statement, without proof, was given in [1, Ch. 4.3].
CAMILLE LAURENT-GENGOUX, EVA MIRANDA , AND POL VANHAECKE (2) The rank of Π at m is r ; (3) The integral manifold F m of X f , . . . , X f s , passing through m , is com-pact.Then there exists R -valued smooth functions ( σ , . . . , σ s ) and R / Z -valuedsmooth functions ( θ , . . . , θ r ) , defined in a neighborhood U of F m such that (1) The functions ( θ , . . . , θ r , σ , . . . , σ s ) define an isomorphism U ≃ T r × B s ; (2) The Poisson structure can be written in terms of these coordinatesas
Π = r X i =1 ∂∂θ i ∧ ∂∂σ i , in particular the functions σ r +1 , . . . , σ s are Casimirs of Π (restrictedto U ); (3) The leaves of the surjective submersion F = ( f , . . . , f s ) are givenby the projection onto the second component T r × B s , in particular,the functions σ , . . . , σ s depend on the functions f , . . . , f s only. The functions θ , . . . , θ r are called angle coordinates, the functions σ ,. . . , σ r are called action coordinates and the remaining functions σ r +1 , . . . , σ s are called transverse coordinates.Our proof of theorem 1.1, consists of several conceptually different steps,which are in 1-1 correspondence with the (a) topological, (b) group theo-retical, (c) geometrical and (d) analytical aspects of the construction of thecoordinates. It parallels Duistermaat’s proof, which deals with the symplec-tic case [7]; while (a) and (b) are direct generalizations of his proof, (c) and(d) are however not.(a) The topological part of the proof amounts to showing that in theneighborhood of the invariant manifold F m , we have locally trivial torusfibration (Paragraph 3.2). Once we have shown that the compact invariantmanifolds are the connected components of the fibers of a submersive map(the map F , restricted to some open subset), the proof of this part is similaras in the symplectic case.(b) The (commuting) Hamiltonian vector fields are tangent to the toriof this fibration; integrating them we get an induced torus action (actionby T r ) on each of these tori, but in general these actions cannot be com-bined into a single torus action. Taking appropriate linear combinations ofthe vector fields, using F -basic functions as coefficients, by a procedure called“uniformization of the periods”, one constructs new vector fields Y , . . . , Y r which are tangent to the fibration, and which now integrate into a singletorus action. This is the content of step 1 in the proof of proposition 3.6.This step, which is an application of the implicit function theorem, is iden-tical as in the symplectic case. CTION-ANGLE COORDINATES 5 (c) The newly constructed vector fields Y i are the fundamental vectorfields of a torus action. We first show that they are Poisson vector fields,i.e., that they preserve the Poisson structure (step 2 in the proof of propo-sition 3.6). The key (and non-trivial) point of the proof is the periodicityof the vector fields Y i . We then prove (in step 3 of the proposition) thestronger statement that these vector fields are Hamiltonian vector fields, atleast locally, by constructing quite explicitly their Hamiltonians, which willin the end play the role of action coordinates.(d) In the last step (theorem 3.8), we use the Carath´eodory-Jacobi-Lietheorem for Poisson manifolds to construct on the one hand coordinateswhich are conjugate to the action coordinates (angle coordinates and trans-verse coordinates) and on the other hand to extend these coordinates to aneighborhood of the Liouville torus F m . The Carath´eodory-Jacobi-Lie the-orem for Poisson manifolds, to which Section 2 is entirely devoted, providesa set of canonical local coordinates for a Poisson structure Π, containing agiven set p , . . . , p r of functions in involution. It generalizes both the classicalCarath´eodory-Jacobi-Lie theorem for symplectic manifolds [12, Th. 13.4.1]and Weinstein’s splitting theorem [19, Th. 2.1]. We are convinced that thistheorem, which is new, has other interesting applications, as in the study oflocal forms and stability of integrable systems.The action-angle theorem has been proven by [17] in the general context ofnon-commutative integrable systems on a symplectic manifold (see the Ap-pendix for a comparison between this notion and some closely related notionsof integrability). Roughly speaking, a non-commutative integrable systemhas more constants of motion than a Liouville integrable system, accountingfor linear motion on smaller tori, but not all these functions are in involu-tion. This notion has a natural definition in the case of Poisson manifolds,proposed here (definition 4.1); it generalizes both the notion of Liouvilleintegrability on a Poisson manifold and the notion of non-commutative inte-grability on a symplectic manifold. We show in Section 4 that our proof canbe adapted (i.e., generalized) to provide a proof of the action-angle theoremin this very general context.The structure of the paper is as follows. We state and prove the Carath´eo-dory-Jacobi-Lie theorem for Poisson manifolds in Paragraph 2.1 and we givein Paragraph 2.2 a counterexample which shows that a mild generalizationof the latter theorem does not hold in general. The action-angle theoremfor Liouville integrable systems on Poisson manifolds is given in Section3. We show in Section 4 how this theorem can be adapted to the moregeneral case of non-commutative integrable systems on Poisson manifolds.The appendix to the paper is devoted to the geometrical formulation of thenotion of a non-commutative integrable system on a Poisson manifold.In this paper, all manifolds and objects considered on them are smoothand we write { f, g } for Π(d f, d g ). CAMILLE LAURENT-GENGOUX, EVA MIRANDA , AND POL VANHAECKE The Carath´eodory-Jacobi-Lie theorem for Poisson manifolds
In this section we prove a natural generalization of the classical Carath´eo-dory-Jacobi-Lie theorem [12, Th. 13.4.1] for an arbitrary Poisson manifold( M, Π). It provides a set of canonical local coordinates for the Poissonstructure Π, which contains a given set p , . . . , p r of functions in involution(i.e., functions which pairwise commute for the Poisson bracket), whoseHamiltonian vector fields are assumed to be independent at a point m ∈ M (theorem 2.1). This result, which is interesting in its own right, will be usedin our proof of the action-angle theorem. We show in Paragraph 2.2 bygiving a counterexample that canonical coordinates containing a given setof functions in involution may fail to exist as soon as the Hamiltonian vectorfields X p , . . . , X p r are dependent at m , even if they are independent at allother points in a neighborhood of m .2.1. The theorem.
The main result of this section is the following theorem.
Theorem 2.1.
Let m be a point of a Poisson manifold ( M, Π) of dimen-sion n . Let p , . . . , p r be r functions in involution, defined on a neighbor-hood of m , which vanish at m and whose Hamiltonian vector fields are lin-early independent at m . There exist, on a neighborhood U of m , functions q , . . . , q r , z , . . . , z n − r , such that (1) The n functions ( p , q , . . . , p r , q r , z , . . . , z n − r ) form a system ofcoordinates on U , centered at m ; (2) The Poisson structure Π is given on U by Π = r X i =1 ∂∂q i ∧ ∂∂p i + n − r X i,j =1 g ij ( z ) ∂∂z i ∧ ∂∂z j , (2.1) where each function g ij ( z ) is a smooth function on U and is inde-pendent of p , . . . , p r , q , . . . , q r .The rank of Π at m is r if and only if all the functions g ij ( z ) vanish for z = 0 .Proof. We show the first part of the theorem by induction on r . For r = 0,every system of coordinates z , . . . , z n , centered at m , does the job. Assumethat the result holds true for every point in every Poisson manifold and every( r − r >
1. We prove it for r . To dothis, we consider an arbitrary point m in an n -dimensional Poisson manifold( M, Π), and we assume that we are given functions in involution p , . . . , p r ,defined on a neighborhood of m , which vanish at m , and whose Hamiltonianvector fields are linearly independent at m . On a neighbourhood of m , thedistribution D := hX p , . . . , X p r i has constant rank r and is an involutivedistribution because [ X p i , X p j ] = −X { p i ,p j } = 0. By the Frobenius theorem,there exist local coordinates g , . . . , g n , centered at m , such that X p i = ∂∂g i CTION-ANGLE COORDINATES 7 for i = 1 , . . . , r , on a neighbourhood of m . Setting q r := g r we have X q r [ p i ] = −X p i [ q r ] = − δ i,r , i = 1 , . . . , r, (2.2)in particular (1) the r + 1 vectors d m p , . . . , d m p r and d m q r of T ∗ m M arelinearly independent, and (2) the vector fields X q r and X p r are independentat m . It follows that a distribution D ′ (of rank 2) is defined by X q r and X p r .It is an integrable distribution because [ X q r , X p r ] = −X { q r ,p r } = 0 . Applied to D ′ , the Frobenius theorem yields the existence of local coordinates v , . . . , v n ,centered at m , such that X p r = ∂∂v n − and X q r = ∂∂v n . (2.3)Since the differentials d m v , . . . , d m v n − vanish on X p r ( m ) and on X q r ( m ), itfollows that (d m v , . . . , d m v n − , d m p r , d m q r ) is a basis of T ∗ m M . Therefore,the n functions ( v , . . . , v n − , p r , q r ) form a system of local coordinates, cen-tered at m . It follows from (2.3) that the Poisson structure takes in termsof these coordinates the following form:Π = ∂∂q r ∧ ∂∂p r + n − X i,j =1 h ij ( v , . . . , v n − , p r , q r ) ∂∂v i ∧ ∂∂v j . The Jacobi identity, applied to the triplets ( p r , v i , v j ) and ( q r , v i , v j ), impliesthat the functions h ij do not depend on the variables p r , q r , so thatΠ = ∂∂q r ∧ ∂∂p r + n − X i,j =1 h ij ( v , . . . , v n − ) ∂∂v i ∧ ∂∂v j , (2.4)which means that Π is, in a neighborhood of m , the product of a symplecticstructure (on a neighborhood of the origin in R ) and a Poisson structure(on a neighborhood of the origin in R n − ). In order to apply the recursionhypothesis, we need to show in case r − > p , . . . , p r − depend onlyon the coordinates v , . . . , v n − , i.e., are independent of p r and q r , ∂p i ∂p r = 0 = ∂p i ∂q r i = 1 , . . . , r − . (2.5)Both equalities in (2.5) follow from the fact that p i is in involution with p r and q r , for i = 1 , . . . , r −
1, combined with (2.4):0 = { p i , p r } = ∂p i ∂q r , { p i , q r } = − ∂p i ∂p r . We may now apply the recursion hypothesis on the second term in (2.4),together with the functions p , . . . , p r − . It leads to a system of local coor-dinates ( p , q , . . . , p r , q r , z , . . . , z n − r ) in which Π is given by (2.1). Thisshows the first part of the theorem. The second part of the theorem is aneasy consequence of (2.1), since it implies that the rank of Π at m is 2 r plusthe rank of the second term in the right hand side of (2.1), at z = 0. (cid:3) CAMILLE LAURENT-GENGOUX, EVA MIRANDA , AND POL VANHAECKE Remark . The classical Carath´eodory-Jacobi-Lie theorem corresponds tothe case dim M = 2 r . Then Π is the Poisson structure associated to asymplectic structure, in the neighborhood of m . Theorem 2.1 then saysthat Π can be written in the simple formΠ = r X i =1 ∂∂q i ∧ ∂∂p i , (2.6)where we recall that the (involutive) set of functions p , . . . , p r is prescribed. Remark . Theorem 2.1 and their proof, as they are stated, do not yield theexistence of the involutive set of functions p , . . . , p r , a fact which is plainin Weinstein’s splitting theorem. However, if we forget in our proof thatthese functions are prescribed, we can easily adapt the induction hypotheses,adding the existence of r such functions, when the rank of the Poissonstructure at m is at least 2 r . In this sense, our theorem is an amplificationof Weinstein’s splitting theorem. Remark . Theorem 2.1 holds true for holomorphic Poisson manifolds; thelocal coordinates are in this case holomorphic coordinates and the functions g ij ( z ) are holomorphic functions, independent from p , . . . , p r , q , . . . , q r . Upto these substitutions, the given proof is valid word by word.2.2. A counterexample.
If we denote in theorem 2.1 the rank of Π at m by 2 r ′ , then 2 r ′ > r , because the involutive set of functions p , . . . , p r define a totally isotropic foliation in a neighborhood of m . It means that,if 2 r ′ < r and one is given independent functions in involution p , . . . , p r ,then their Hamiltonian vector fields X p , . . . , X p r are dependent at m . In theextremal case in which dim hX p ( m ) , . . . , X p r ( m ) i = r ′ one has , accordingto theorem 2.1, that there exist functions q , . . . , q r ′ and z , . . . , z n − r ′ suchthat Π takes the formΠ = r ′ X i =1 ∂∂q i ∧ ∂∂p i + n − r ′ X k,l =1 φ k,l ( z , . . . , z n − r ′ ) ∂∂z k ∧ ∂∂z l . A natural question is whether r − r ′ of the functions z i can be chosenas p r ′ +1 , . . . , p r , or, more generally, as functions which depend only on p , . . . , p r . We show in the following (counter) example that this is notpossible, in general. Example . On R , with coordinates f , f , g , g , consider the bivectorfield, given byΠ = ∂∂g ∧ ∂∂f + χ ( g ) ∂∂g ∧ ∂∂f + ψ ( g ) ∂∂g ∧ ∂∂f , (2.7)where χ ( g ) and ψ ( g ) are smooth functions that depend only on g , andwhich vanish for g = 0, so that the rank of Π at the origin is 2. A direct Possibly up to a relabelling of the p i , so that dim ˙ X p ( m ) , . . . , X p r ′ ( m ) ¸ = r ′ . CTION-ANGLE COORDINATES 9 computation shows that this bivector field is a Poisson bivector field andthat f and f are in involution. We show that for some choice of χ and ψ there exists no system of coordinates p , q , z , z , centered at 0, with p , z depending only on f and f , such thatΠ = ∂∂q ∧ ∂∂p + φ ( z , z ) ∂∂z ∧ ∂∂z . (2.8)To do this, let us assume that such a system of coordinates exists. Takingthe Poisson bracket of p = p ( f , f ) and z = z ( f , f ) with q yields, inview of (2.8), 1 = { q , p } = ∂p ∂f { q , f } + ∂p ∂f { q , f } , { q , z } = ∂z ∂f { q , f } + ∂z ∂f { q , f } . (2.9)Let N denote the locus defined by f = f = 0, which is a smooth surface ina neighborhood of the origin. Let q denote the restriction of q to N . Since X f and X f are tangent to N , X f i [ q ] = { q , f i } | N , so that (2.9), restrictedto N , becomes 1 = λ X f [ q ] + λ X f [ q ] , λ X f [ q ] + λ X f [ q ] , (2.10)where λ , . . . , λ are constants (because p , z depend only on f , f ), andsatisfy λ λ − λ λ = 0, since p and z are part of a coordinate systemcentered at the origin. It follows that X f [ q ] = c and X f [ q ] = c , (2.11)where c and c are constants, which cannot be both equal to zero, in viewof (2.10). Writing X f and X f in terms of the original variables, using (2.7),we find that q = q ( g , g ) must satisfy ∂q∂g = c , χ ( g ) ∂q∂g + ψ ( g ) ∂q∂g = c . Evaluating the second equation at g = g = 0 gives c = 0, hence c = 0and q ( g , g ) = c g + r ( g ) for some smooth function r ( g ). Then the secondcondition leads to the following differential equation for r , χ ( g ) r ′ ( g ) = − ψ ( g ) c . (2.12)But this equation does not admit a smooth solution, unless ψ ( g ) /χ ( g ) ad-mits a smooth continuation at 0. If, for example, ψ ( g ) = g and χ ( g ) = g ,then there is no solution r ( g ) to (2.12), which is smooth in the neighbor-hood of 0, hence a system of coordinates in which Π takes the form (2.8)does not exist. , AND POL VANHAECKE Action-angle coordinates for Liouville integrable systemson Poisson manifolds
In this section we prove the existence of action-angle coordinates in theneighborhood of every standard Liouville torus of an integrable system onan arbitrary Poisson manifold.3.1.
Standard Liouville tori of Liouville integrable systems.
We firstrecall the definition of a Liouville integrable system on a Poisson manifold.
Definition 3.1.
Let ( M, Π) be a Poisson manifold of (maximal) rank 2 r and of dimension n . An s -tuplet of functions F = ( f , . . . , f s ) on M is saidto define a Liouville integrable system on ( M, Π) if(1) f , . . . , f s are independent (i.e., their differentials are independenton a dense open subset of M );(2) f , . . . , f s are in involution (pairwise);(3) r + s = n .Viewed as a map, F : M → R s is called the momentum map of ( M, Π , F ).We denote by M r the open subset of M where the rank of Π is equal to 2 r ;points of M r are called regular points of M . We denote by U F the denseopen subset of M , which consists of all points of M where the differentialsof the elements of F are linearly independent, U F := { m ∈ M | d m f ∧ d m f ∧ . . . ∧ d m f s = 0 } . (3.1)On the non-empty open subset M r ∩ U F of M the Hamiltonian vector fields X f , . . . , X f s define a distribution D of rank r , since at each point m of M r the kernel of Π m has dimension n − r = s − r . The distribution D isintegrable because the vector fields X f , . . . , X f s pairwise commute, (cid:2) X f i , X f j (cid:3) = −X { f i ,f j } = 0 , for 1 i < j s . The integral manifolds of D are the leaves of a regularfoliation, which we denote by F ; the leaf of F , passing through m , is denotedby F m , and is called the invariant manifold of F , through m . For whatfollows, we will be uniquely interested in the case in which F m is compact.According to the classical Liouville theorem, adapted to the case of Poissonmanifolds (see [1, Sect. 4.3] for a proof in the Poisson manifold case), everycompact invariant manifold F m is diffeomorphic to the torus T r := ( R / Z ) r ;more precisely, the diffeomorphism can be chosen such that each of thevector fields X f i is sent to a constant (i.e., translation invariant) vector fieldon T r . Such a torus is called a standard Liouville torus .3.2. Foliation by standard Liouville tori.
As a first step in establishingthe existence of action-angle coordinates, we prove that, in some neighbor-hood of a standard Liouville torus, the invariant manifolds of an integrablesystem ( M, Π , F ) form a trivial torus fibration. CTION-ANGLE COORDINATES 11
Proposition 3.2.
Suppose that F m is a standard Liouville torus of an in-tegrable system ( M, Π , F ) of dimension n := dim M and rank r := Rk Π .There exists an open subset U ⊂ M r ∩ U F , containing F m , and there exists adiffeomorphism φ : U ≃ T r × B n − r , which takes the foliation F to the folia-tion, defined by the fibers of the canonical projection p B : T r × B n − r → B n − r ,leading to the following commutative diagram. F m U T r × B n − r B n − r (cid:31) (cid:127) / / / / φ / / ≃ (cid:15) (cid:15) F | U z z tttttttttt p B Proof.
We first show that the foliation F , which consists of the maximalintegral manifolds of the foliation D , defined by the integrable vector fields X f , . . . , X f s , where s := n − r , coincides with the foliation ¯ F , defined by thefibers of the submersion¯ F = ( f , . . . , f s ) : M r ∩ U F → R s , which is the restriction of F : M → R s to M r ∩ U F . Since all leaves of ¯ F and of F are r -dimensional, it suffices to show that the two leaves, whichpass through an arbitrary point m ∈ M r ∩ U F , have the same tangent spaceat m . Since f , . . . , f s are pairwise in involution, each of the vector fields X f , . . . , X f s is tangent to the fibers of ¯ F , i.e., to the leaves of ¯ F . Thus, T m F ⊂ T m ¯ F , which implies that both tangent spaces are equal, since theyhave the same dimension r .Suppose now that F m is a standard Liouville torus. We show that thereexists a neighborhood U of F m and a diffeomorphism φ : U → F m × B s ,which sends the foliation ¯ F (= F ), restricted to U , to the foliation definedby p B on F m × B s . The proof of this fact depends only on the fact that F m is a compact component of a fiber of a submersion (namely ¯ F ). Notice thatsince ¯ F is a submersion, every point m ′ ∈ ¯ F m = F m has a neighborhood U m ′ in M , which is diffeomorphic to the product of a neighborhood V m ′ of m ′ in F m times an open ball B sm ′ , centered at ¯ F ( m ′ ) = ¯ F ( m ) in R s ; sucha diffeomorphism φ m ′ , as provided by the implicit function theorem, is alifting of ¯ F , i.e., it leads to the following commutative diagram: U m ′ V m ′ × B sm ′ B sm ′ / / ___ φ m ′ $ $ JJJJJJJJJJJ ¯ F (cid:15) (cid:15) p B Since F m is compact, it is covered by finitely many of the sets V m ′ , say V m , . . . , V m ℓ . Thus, if every pair of the diffeomorphisms φ m , . . . , φ m ℓ agreeson the intersection of their domain of definition (whenever non-empty), we , AND POL VANHAECKE can define a global diffeomorphism on a neighborhood U of F m , whose imageis the intersection of the concentric balls B sm , . . . , B sm ℓ . In order to ensurethat these diffeomorphisms agree, we need to chose them in a more specificway. This is done by choosing an arbitrary Riemannian metric on M . Usingthe exponential map, defined by the metric, we can identify a neighborhoodof the zero section in the normal bundle of F m , with a neighborhood of F m in M ; in particular, for every m ′ ∈ F m there exist neighborhoods U m ′ of m ′ in M and V m ′ of m ′ in F m , with smooth maps ψ m ′ : U m ′ → V m ′ ,which have the important virtue that they agree on the intersection of theirdomains. Upon shrinking the open subsets U m ′ , if necessary, the maps φ m ′ := ψ m ′ × ( f , . . . , f s ) are a choice of diffeomorphisms, defined on aneighborhood U of F m , with the required properties. (cid:3) Corollary 3.3.
Suppose that F m is a standard Liouville torus of an inte-grable system ( M, Π , F ) of dimension n := dim M and rank r := Rk Π .There exists an open subset U ⊂ M r ∩ U F , containing F m , and there exist n − r functions z , . . . , z n − r on U which are Casimir functions of Π , andwhose differentials are independent at every point of U .Proof. Let U ⊂ M r ∩ U F and φ be as given by proposition 3.2. We consider,besides D , another integrable distribution on U : the distribution D ′ definedby all Hamiltonian vector fields on U ; it has rank 2 r and its leaves arethe symplectic leaves of ( U, Π). Since D is the distribution, defined by theHamiltonian vector fields X f , . . . , X f s , we have that D ⊂ D ′ . Considerthe submersive map p B ◦ φ : U → T r × B s → B s , whose fibers are byassumption the leaves of F , i.e., the integral manifolds of D (restrictedto U ), so that the kernel of d( p B ◦ φ ) is precisely D . The image of D ′ byd( p B ◦ φ ) is therefore a (smooth) distribution D ′′ of rank r on B s , whichis integrable, since D ′ is integrable. The foliation defined by the integralmanifolds of D ′′ is, in the neighborhood of the point p B ( φ ( m )), defined by s − r = n − r independent functions z ′ , . . . , z ′ n − r . Pulling them back to M ,we get functions z , . . . , z n − r on a neighborhood U of F m , with independentdifferentials on U , and they are Casimir functions because they are constanton the leaves of D ′ , which are the symplectic leaves of ( U, Π). (cid:3)
For Liouville tori in an integrable system, which are not standard, theremay not exist a neighborhood on which the invariant manifolds of the inte-grable system are locally trivial. We show this in the following example.
Example . Let M be the product of a M¨obius band with an interval,which is obtained by identifying on M := [ − , × ] − , × R in pairs thepoints ( − , y, z ) and (1 , − y, z ), where y and z are arbitrary. On M , considerthe vector field V := ∂/∂x , the Poisson structureΠ := ∂∂x ∧ ∂∂z , CTION-ANGLE COORDINATES 13 and the function F := z . The algebra of Casimir functions of Π consists ofall smooth functions on M that are independent of x and z (i.e., arbitrarysmooth functions in y ). Clearly, both V and Π and z go down to M ,yielding a vector field V , a Poisson structure Π = V ∧ ∂/∂z and a function z on M . What does not go down to M is the function y . In fact, onlyeven functions in y go down and the algebra of Casimir functions of Π is thealgebra of even functions in y , viewed as functions on M . This remains trueif we restrict M to any neighborhood of the central circle y = z = 0, whichis a leaf of the foliation, defined by the fibers of F . Since the differential ofan even function in y vanishes at all points where y = 0, the central circleis not a standard Liouville torus. Since every neighborhood of the centralcircle contains leafs that spin around the M¨obius band twice, the Liouvilletori do not form a locally trivial torus fibration in the neighborhood of thecentral circle.3.3. Standard Liouville tori and Hamiltonian actions.
According toproposition 3.2, the study of an integrable system ( M, Π , F ) in the neigh-borhood of a standard Liouville torus amounts to the study of an integrablesystem ( T r × B n − r , Π , p B ), where Π is a Poisson structure on T r × B n − r of constant rank 2 r and the map p B : T r × B n − r → B n − r is the projectionon the second factor. We write the latter integrable system in the sequelas ( T r × B s , Π , F ) and we denote the components of F by F = ( f , . . . , f s )where s := n − r , as before. We show in the following lemma that we mayassume that the first r vector fields X f , . . . , X f r are independent on T r × B s ,hence span the fibers of F at each point. Lemma 3.5.
Let ( T r × B s , Π , F ) be an integrable system, where Π hasconstant rank r and F : T r × B s → B s denotes the projection on the secondcomponent. Let m ∈ T r × { } and suppose that the components of F =( f , . . . , f s ) are ordered such that the Hamiltonian vector fields X f , . . . , X f r are independent at m . There exists a ball B s ⊂ B s , centered at , such that X f , . . . , X f r are independent on T r × B s .Proof. We denote by L V the Lie derivative with respect to a vector field V .Since the vector fields X f i pairwise commute, L X fj ( X f ∧ . . . ∧ X f r ) = r X i =1 X f ∧ . . . ∧ (cid:2) X f j , X f i (cid:3) ∧ . . . ∧ X f r = 0 , for j = 1 , . . . , s . It means that X f ∧ . . . ∧ X f r is conserved by the flowof each one of the vector fields X f , . . . , X f s . In particular, if this r -vectorfield is non-vanishing at m ∈ T r × { } then it is non-vanishing on theentire integral manifold through m of the distribution D , defined by thesevector fields. Since this integral manifold, which is a torus, is compact, it isactually non-vanishing on a neighborhood of the integral manifold, which wecan choose of the form T r × B s , where B s ⊂ B s is a ball, centered at 0. (cid:3) , AND POL VANHAECKE Given an integrable system ( T r × B s , Π , F ), where Π has constant rankand F = ( f , . . . , f s ) is the projection on the second component, the Hamil-tonian vector fields X f i need not be constant on the fibers of F (which aretori), and even if they are, they may vary from one fiber to another inthe sense that they do not come from the single action of the torus T r on T r × B s . We show in the following proposition how this can be achieved,upon replacing the Hamiltonian vector fields X f i by well-chosen linear com-binations, with as coefficients F -basic functions, i.e., functions of the form F ◦ λ , where λ ∈ C ∞ ( B s ); equivalently, smooth functions on T r × B s whichare constant on the fibers of F . Proposition 3.6.
Let ( T r × B s , Π , F ) be an integrable system, where Π hasconstant rank r and F = ( f , . . . , f s ) is projection on the second component.Suppose that the r vector fields X f , . . . , X f r are independent at all points of T r × B s . There exists a ball B s ⊂ B s , also centered at , and there exist F -basic functions λ ji ∈ C ∞ ( B s ) , such that the r vector fields Y i := P rj =1 λ ji X f j , ( i = 1 , . . . , r ) , are the fundamental vector fields of a Hamiltonian torusaction of T r on T r × B s . The proof uses the following lemma.
Lemma 3.7.
Let Y be a Poisson vector field on a Poisson manifold ( M, Π) of dimension n and rank r . If Y is tangent to all symplectic leaves of M ,then Y is Hamiltonian in the neighborhood of every point m ∈ M where therank of Π is r .Proof. If the rank of Π at m is 2 r , so that m is a regular point of Π, then thereexists local coordinates ( p , q , . . . , p r , q r , z , . . . , z n − r ) in a neighborhood U of m with respect to which the Poisson structure P is given by:Π = r X i =1 ∂∂q i ∧ ∂∂p i . The vector fields ∂∂q , ∂∂p , . . . , ∂∂q r , ∂∂p r span the symplectic leaves of Π on U .Therefore, every vector field Y , which is tangent to the symplectic leaves ofΠ, is of the form Y = r X i =1 a i ∂∂p i + r X i =1 b i ∂∂q i for some smooth functions a , . . . , a r , b , . . . b r , defined on U . The relation[ Y , Π] = 0 imposes the following set of equations to be satisfied for all i, j = 1 , . . . , r : ∂a i ∂q j = ∂a j ∂q i , ∂b i ∂p j = ∂b j ∂p i and ∂a i ∂p j = − ∂b i ∂q j CTION-ANGLE COORDINATES 15
By the classical Poincar´e lemma, there exists a function h , defined on U ,which satisfies, for i = 1 , . . . , r : a i = − ∂h∂q i and b i = ∂h∂p i . Hence, X h = r X i =1 ∂h∂q i X q i + r X i =1 ∂h∂p i X p i + n − r X k =1 ∂h∂z k X z k = Y , which shows that Y is a Hamiltonian vector field on U . (cid:3) Now, we can turn our attention to the proof of proposition 3.6.
Proof.
The fibers of F = ( f , . . . , f s ) are compact, so for i = 1 , . . . , r , theflow Φ ( i ) t i of the Hamiltonian vector field X f i is complete and we can definea map, Φ : R r × ( T r × B s ) → T r × B s (( t , . . . , t r ) , m ) Φ (1) t ◦ · · · ◦ Φ ( r ) t r ( m ) . Since the vector fields X f i are pairwise commuting, the flows Φ ( i ) t i pairwisecommute and Φ is an action of R r on T r × B s . Since the vector fields X f , . . . , X f r are independent at all points, the fibers of F , which are r -dimensional tori, are the orbits of the action. For c ∈ B s , let Λ c denotethe lattice of R r , which is the isotropy group of any point in F − ( c ); it isthe period lattice of the action Φ, restricted to F − ( c ). Notice that if Λ c isindependent of c ∈ B s , the action Φ descends to an action of T r = R r / Λ c on T r × B s . We will show in Step 1 below that this independence can be assuredafter applying a diffeomorphism of T r × B s over B s , where B s is a ball,contained in B s , and concentric with it. The proof of this step is essentiallythe same as in the symplectic case; it is called uniformization of the periods .Steps 2 and 3 below prove successively that the fundamental vector fieldsof the obtained torus action are Poisson, respectively Hamiltonian vectorfields.Step 1. The periods of Φ can be uniformized to obtain a torus action of T r on T r × B s , whose orbits are the fibers of F (restricted to T r × B s ).Let m be an arbitrary point of F − (0) and choose a basis ( λ (0) , . . . , λ r (0))for the lattice Λ . For a fixed i , with 1 i r , for m in a neighborhoodof m in T r × B s and for L in a neighborhood of λ i (0) in R r , consider theequation Φ( L, m ) = m . Since F (Φ( L, m )) = F ( m ) for all L and m , it ismeaningful to write Φ( L, m ) − m and solving the equation Φ( L, m ) = m locally for L amounts to applying the implicit function theorem to the map R r × ( T r × B s ) T r × B s T r . / / Φ( L,m ) − m / / Since the action is locally free, the Jacobian condition is satisfied and we getby solving for L around λ i (0) a smooth R r -valued function λ i ( m ), definedfor m in a neighborhood W i of m . Doing this for i = 1 , . . . , r and setting , AND POL VANHAECKE W := ∩ ri =1 W i , we have that W is a neighborhood of m , and on W we havefunctions λ ( m ) , . . . , λ r ( m ), with the property that Φ( λ i ( m ) , m ) = m for all m ∈ W and for all 1 i r . Thus, λ ( m ) , . . . , λ r ( m ) belong to the latticeΛ F ( m ) for all m ∈ W and they form a basis when m = m ; by continuity,they form a basis for Λ F ( m ) for all m ∈ W .The functions λ i can be extended to a neighborhood of the torus F − (0).In fact, the functions λ i are F -basic, hence extend uniquely to F -basic func-tions on F − ( F ( W )). We will use in the sequel the same notation λ i for theseextensions and we write F − ( F ( W )) simply as W . Using these functions wedefine the following smooth map:˜Φ : R r × W → W (( t , . . . , t r ) , m ) Φ r X i =1 t i λ i ( m ) , m ! . (3.2)Since the functions λ i are F -basic, the fact that Φ is an action implies that˜Φ is an action. The new action has the extra feature that the stabilizer ofevery point in W is Z r . Thus, ˜Φ induces an action of T r on W , which westill denote by ˜Φ. By shrinking W , if necessary, we may assume that W is of the form F − ( B s ), where B s is an open ball, concentric with B s , andcontained in it. Thus we have a torus action˜Φ : T r × W → W (( t , . . . , t r ) , m ) Φ r X i =1 t i λ i ( m ) , m ! . Step 2. The fundamental vector fields of the torus action ˜Φ are Poissonvector fields.We denote by Y , . . . , Y r the fundamental vector fields of the torus ac-tion ˜Φ, constructed in step 1. We need to show that L Y i Π = 0, or in termsof the Schouten bracket, that [ Y i , Π] = 0, for i = 1 , . . . , r . To do this, we firstexpand Y i in terms of the Hamiltonian vector fields X f , . . . , X f r : since theaction ˜Φ leaves the fibers of F invariant and since the Hamiltonian vectorfields X f , . . . , X f r span the tangent space to these fibers at every point, wecan write Y i = r X j =1 λ ji X f j . (3.3)Since all Hamiltonian vector fields leave Π invariant, L Y i Π = [ Y i , Π] = r X j =1 h λ ji X f j , Π i = r X j =1 X λ ji ∧ X f j , (3.4)which we need to show to be equal to zero. Notice that since the coefficients λ i in the definition of ˜Φ are F -basic, the coefficients λ ji are also F -basic, sothey are pairwise in involution, and their Hamiltonian vector fields commute CTION-ANGLE COORDINATES 17 with all Hamiltonian vector fields X f k . In particular it follows from (3.4)that [ X f k , [ Y i , Π]] = 0 for k = 1 , . . . , r . We derive from it and from (3.3),that [ Y i , [ Y i , Π]] = 0, i.e., that the flow of Y i preserves L Y i Π:[ Y i , [ Y i , Π]] = " r X k =1 λ ki X f k , [ Y i , Π] = r X k =1 h λ ki , [ Y i , Π] i ∧ X f k = r X j,k =1 X f j h λ ki i X f k ∧ X λ ji + r X j,k =1 X λ ji h λ ki i X f k ∧ X f j = 0 , (3.5)since any two F -basic functions are in involution. Hence, L Y i Π = 0. Since Y i is a complete vector field, and has period 1, we can conclude that L Y i Π = 0using the following:Claim. If Y is a complete vector field of period 1 and P is a bivectorfield for which L Y P = 0, then L Y P = 0.In order to prove this claim, we let Q := L Y P and we denote the flowof Y by Φ t . We pick an arbitrary point m and we show that Q m = 0. Wehave for all t that ddt (cid:0) (Φ t ) ∗ P Φ − t ( m ) (cid:1) = (Φ t ) ∗ ( L Y P ) Φ − t ( m ) = (Φ t ) ∗ Q Φ − t ( m ) = Q m , (3.6)where we used in the last step that the bivector field Q satisfies L Y Q = 0.By integrating (3.6), (Φ t ) ∗ P Φ − t ( m ) = P m + tQ m . Evaluated at t = 1 this yields Q m = 0, since Φ = Id, as Y has period 1.Step 3. The vector fields Y , . . . , Y r are Hamiltonian vector fields (withrespect to commuting Hamiltonian functions).According to Step 2, the vector fields Y , . . . , Y r are Poisson vector fields.Since they are tangent to the symplectic leaves, according to lemma 3.7,there is a neighborhood of m ∈ F m in W that we can assume to be of theform Ω r × W s , with Ω r ⊂ T r , W s ⊂ B s , on which the vector fields Y , . . . , Y r are Hamiltonian vector fields. In other words, there exists functions that weshall denote by h , . . . , h r , defined on W s , satisfying the relation Y i = X h i for all i = 1 , . . . , r . It shall be convenient to denote by W again the opensubset F − ( W s ).Let d µ be a Haar measure on T r . For all m ′ ∈ W , we set: U m ′ := { t ∈ T r | ˜Φ t ( m ′ ) ∈ W } As before, Q m denotes the bivector Q at the point m . , AND POL VANHAECKE where, for all t = ( t , . . . , t r ) ∈ T r , ˜Φ t is a shorthand for the map m ′ ˜Φ( t , . . . , t r , m ′ ). We then define functions p i , i = 1 , . . . , r on W by: p i ( m ′ ) := 1vol( U m ′ ) Z t ∈ U m ′ h i (cid:0) ˜Φ t ( m ′ ) (cid:1) d µ where vol( U m ′ ) stands for the volume with respect to the Haar measure.Their Hamiltonian vector fields can be computed as follows,: X p i ( m ′ ) = 1vol( U m ′ ) Z t ∈ U m ′ X h i ◦ ˜Φ t ( m ′ )d µ = 1vol( U m ′ ) Z t ∈ U m ′ d ˜Φ − t (cid:0) X h i ( ˜Φ t ( m ′ )) (cid:1) d µ = 1vol( U m ′ ) Z t ∈ U m ′ d ˜Φ − t (cid:0) Y i ( ˜Φ t ( m ′ )) (cid:1) d µ = 1vol( U m ′ ) Z t ∈ U m ′ Y i ( m ′ )d µ (3.7)= Y i ( m ′ ) , (3.8)where the fact that Y i is invariant under ˜Φ t has been used to go from thethird to the fourth line. The relation U ˜Φ t ′ ( m ′ ) = ˜Φ t ′ ( U m ′ ) for all t ′ ∈ T r ,and the invariance property of the Haar measure, imply that the functions p , . . . , p r are invariant under the T r -action. In particular, they are in invo-lution for all i, j = 1 , . . . , r , since { p i , p j } = Y j [ p i ] = 0 . In conclusion, on the open subset W , the vector fields Y , . . . , Y r are theHamiltonian vector fields of the commuting functions p , . . . , p r . (cid:3) The existence of action-angle coordinates.
We are now ready toformulate and prove the action-angle theorem, for standard Liouville tori inPoisson manifolds.
Theorem 3.8.
Let ( M, Π , F ) be an integrable system, where ( M, Π) is aPoisson manifold of dimension n and rank r . Suppose that F m is a stan-dard Liouville torus, where m ∈ M r ∩U F . Then there exists R -valued smoothfunctions ( p , . . . , p n − r ) and R / Z -valued smooth functions ( θ , . . . , θ r ) , de-fined in a neighborhood U of F m such that (1) The functions ( θ , . . . , θ r , p , . . . , p n − r ) define an isomorphism U ≃ T r × B n − r ; (2) The Poisson structure can be written in terms of these coordinatesas
Π = r X i =1 ∂∂θ i ∧ ∂∂p i , CTION-ANGLE COORDINATES 19 in particular the functions p r +1 , . . . , p n − r are Casimirs of Π (re-stricted to U ); (3) The leaves of the surjective submersion F = ( f , . . . , f n − r ) are givenby the projection onto the second component T r × B n − r , in particular,the functions p , . . . , p n − r depend on the functions f , . . . , f n − r only.The functions θ , . . . , θ r are called angle coordinates , the functions p , . . . , p r are called action coordinates and the remaining coordinates p r +1 , . . . , p n − r are called transverse coordinates .Proof. We denote s := n − r , as before. Since F m is a standard Liouvilletorus, proposition 3.2 and corollary 3.3 imply that there exist on a neigh-borhood U ′ of F m in M on the one hand Casimir functions p r +1 , . . . , p s andon the other hand F -basic functions p , . . . , p r , such that p := ( p , . . . , p s )and F define the same foliation on U ′ , and such that the Hamiltonian vectorfields X p , . . . , X p r are the fundamental vector fields of a T r -action on U ′ ,where each of the vector fields has period 1; the orbits of this torus actionare the leaves of the latter foliation. In view of the Carath´eodory-Jacobi-Lietheorem (theorem 2.1), there exist on a neighborhood U ′′ ⊂ U ′ of m in M , R -valued functions θ , . . . , θ r such thatΠ = r X j =1 ∂∂θ j ∧ ∂∂p j . (3.9)On U ′′ , X p j = ∂∂θ j , for j = 1 , . . . , r ; since each of these vector fields has pe-riod 1 on U ′ , it is natural to view these functions as R / Z -valued functions,which we will do without changing the notation. Notice that the functions θ , . . . , θ r are independent and pairwise in involution on U ′′ , as a trivial con-sequence of (3.9). In particular, θ , . . . , θ r , p , . . . , p s define local coordinateson U ′′ . In these coordinates, the action of T r is given by( t , . . . , t r ) · ( θ , . . . , θ r , p , . . . , p s ) = ( θ + t , . . . , θ r + t r , p , . . . , p s ) , (3.10)so that the functions θ i uniquely extend to smooth R / Z -valued functionssatisfying (3.10), on U := F − ( F ( U ′′ )), which is an open subset of F m in M ;the extended functions are still denoted by θ i . It is clear that { θ i , p j } = δ ji on U , for all i, j = 1 , . . . , r . Combined with the Jacobi identity, this leads to X p k [ { θ i , θ j } ] = {{ θ i , θ j } , p k } = n θ i , δ kj o − n θ j , δ ki o = 0 , which shows that the Poisson brackets { θ i , θ j } are invariant under the T -action; but the latter vanish on U ′′ , hence these brackets vanish on all of U ,and we may conclude that on U , the functions ( θ , . . . , θ r , p , . . . , p s ) haveindependent differentials, so they define a diffeomorphism to T r × B s where B s is a (small) ball with center 0, and that the Poisson structure takes interms of these coordinates the canonical form (3.9), as required. (cid:3) , AND POL VANHAECKE The results of the present section can be applied in particular for a well-known integrable system constructed on a regular coadjoint orbit O of u ( n ) ∗ ,namely the Gelfand-Cetlin integrable system, for which action-angle coor-dinates are computed explicitly in [11] and [10]. This system can be seenin the Poisson setting, as follows. Dualizing the increasing sequence of Liealgebra inclusions: u (1) ⊂ · · · ⊂ u ( n − ⊂ u ( n )(where u ( k ) is considered as the left-upper diagonal block of u ( k + 1) for k = 1 , . . . , n − u ( n ) ∗ u ( n − ∗ · · · u (1) ∗ / / / / / / / / / / / / The family of functions on u ( n ) ∗ obtained by pulling-back generators of theCasimir algebras of all the u ( k ) ∗ for k = 1 , . . . , n yields a Liouville integrablesystem on u ( n ) ∗ . For particular generators, its restriction to an open subsetof O gives the Gelfand-Cetlin system. The invariant manifold is compact, sothat theorem 3.8 can be applied and gives the existence of action-angle coor-dinates, defined not only in a neighborhood of the invariant manifold in O ,but in a neighborhood of the invariant manifold in the ambient space u ( n ).The restriction of these action and angle coordinates to one symplectic leaf O will give action-angle coordinates on O , as in [11] or [10].4. Action-angle coordinates for non-commutative integrablesystems on Poisson manifolds
In this section, we prove the existence of action-angle coordinates in aneighborhood of a compact invariant manifold in the very general contextof non-commutative integrable systems.4.1.
Non-commutative integrable systems.
We first define preciselywhat we mean by a non-commutative integrable system on a Poisson mani-fold, since the definitions in the literature [3, 9, 8] are only given in the caseof a symplectic manifold. See the appendix for a more intrinsic version ofthis definition.
Definition 4.1.
Let ( M, Π) be a Poisson manifold. An s -tuple of functions F = ( f , . . . , f s ) is said to be a non-commutative integrable system of rank r on ( M, Π) if(1) f , . . . , f s are independent (i.e. their differentials are independent ona dense open subset of M );(2) The functions f , . . . , f r are in involution with the functions f , . . . , f s ;(3) r + s = dim M ;(4) The Hamiltonian vector fields of the functions f , . . . , f r are linearlyindependent at some point of M . CTION-ANGLE COORDINATES 21
We denote the subset of M where the differentials d f , . . . , d f s (resp. wherethe Hamiltonian vector fields X f , . . . , X f r ) are independent by U F (resp. by M F ,r ). Notice that 2 r Rk Π, as a consequence of (4).If ( M, Π , F ) is a Liouville integrable system (definition 3.1), then it is clearthat the components ( f , . . . , f s ) of F can be ordered such that F is a non-commutative integrable system of rank Rk Π. Thus, the notion of a non-commutative integrable system on a Poisson manifold ( M, Π) generalizesthe notion of a Liouville integrable system on ( M, Π). For simplicity, weoften refer in this section to the case of a Liouville integrable system as the commutative case .4.2.
Standard Liouville tori for non-commutative integrable sys-tems.
Let F be a non-commutative integrable system of rank r on a Pois-son manifold ( M, Π) of dimension n . The open subsets U F and M F ,r arepreserved by the flow of each of the vector fields X f , . . . , X f r since eachof the functions f , . . . , f r is in involution with all the functions f , . . . , f s .On the non-empty open subset M F ,r ∩ U F of M , the Hamiltonian vectorfields X f , . . . , X f r define a (regular) distribution D of rank r . Since the vec-tor fields X f , . . . , X f r commute pairwise, the distribution D is integrable,and its integral manifolds are the leaves of a (regular) foliation F . Theleaf through m ∈ M is denoted by F m , and called the invariant manifoldthrough m of F . As in the commutative case, we are only interested inthe case where F m is compact. Under this assumption, F m is a compact r -dimensional manifold, equipped with r independent commuting vector fields,hence it is diffeomorphic to an r -dimensional torus T r ; then F m is called a standard Liouville torus of F . Proposition 3.2 takes in the general situationof a non-commutative integrable system formally the same form, but withthe understanding that r now stands for the rank of F (rather than half therank of the Poisson structure), as stated in the following proposition . Proposition 4.2.
Suppose that F m is a standard Liouville torus of a non-commutative integrable system F of rank r on an n -dimensional Poissonmanifold ( M, Π) . There exists an open subset U ⊂ M F ,r ∩ U F , containing F m , and there exists a diffeomorphism φ : U ≃ T r × B n − r , which takes thefoliation F to the foliation, defined by the fibers of the canonical projection p B : T r × B n − r → B n − r , leading to the following commutative diagram. F m U T r × B n − r B n − r (cid:31) (cid:127) / / / / φ / / ≃ (cid:15) (cid:15) F | U z z tttttttttt p B Recall that B n − r is a ball of dimension n − r . , AND POL VANHAECKE Standard Liouville tori and Hamiltonian actions.
According toproposition 4.2, the study of a non-commutative integrable system ( M, Π , F )of rank r in the neighborhood of a standard Liouville torus amounts tothe study of the non-commutative integrable system ( T r × B n − r , Π , p B )of rank r , where Π is a Poisson structure on T r × B n − r and the map p B : T r × B n − r → B n − r is the projection onto the second factor. We writethe latter integrable system in the sequel as ( T r × B s , Π , F ) and we denotethe components of F by F = ( f , . . . , f s ) where s := n − r , as before. Wemay assume that the first r vector fields X f , . . . , X f r are independent on T r × B s , as shown in the following lemma, the proof of which goes alongthe same lines as the proof of lemma 3.5. Lemma 4.3.
Let ( T r × B s , Π , F ) be a non-commutative integrable systemof rank r , where F : T r × B s → B s denotes the projection onto the secondcomponent. Let m ∈ T r × { } and suppose that the Hamiltonian vectorfields X f , . . . , X f r are independent at m . There exists a ball B s ⊂ B s ,centered at , such that X f , . . . , X f r are linearly independent at every pointof T r × B s . One useful consequence of the fact that the Hamiltonian vector fields X f , . . . , X f r are independent on M := T r × B s is that a function g ∈ C ∞ ( M )is F -basic if and only if X f i [ g ] = 0 for i = 1 , . . . , r . Indeed, g is F -basic ifand only g is constant on all fibers of F , and all tangent spaces to thesefibers are spanned by the vector fields X f , . . . , X f r .We now come to an important difference between the commutative andthe non-commutative case, which is related to the nature of the map F . Inthe commutative case, two F -basic functions on T r × B s are in involution, { g ◦ F , h ◦ F } = 0 for all g, h ∈ C ∞ ( B s ). Said differently, F : ( T r × B s , Π) → ( B s , , is a Poisson map, where B s is equipped with the trivial Poisson structure.The generalization to the non-commutative case is that B s admits a Poissonstructure (non-zero in general), such that F is a Poisson map. This Poissonstructure is constructed by the following (classical) trick: for every pairof functions g, h ∈ C ∞ ( B s ) we have in view of the Jacobi identify, for all i = 1 , . . . , r , X f i [ { g ◦ F , h ◦ F } ] = {X f i [ g ◦ F ] , h ◦ F } + { g ◦ F , X f i [ h ◦ F ] } = 0 , so that { g ◦ F , h ◦ F } is F -basic, namely { g ◦ F , h ◦ F } = { g, h } B ◦ F forsome function { g, h } B ∈ C ∞ ( B s ). It is clear that this defines a Poissonstructure Π B = {· , ·} B on B s and that F : ( T r × B s , Π) → ( B s , Π B )is a Poisson map. This Poisson structure leads to a special class of F -basic functions, which play an important role in the non-commutative case,defined as follows. CTION-ANGLE COORDINATES 23
Definition 4.4.
A smooth function h on T r × B s is said to be a Casimir-basic function , or simply a
Cas-basic function if there exists a Casimir func-tion g on ( B s , Π B ), such that h = g ◦ F .A characterization and the main properties of Cas-basic functions aregiven in the following proposition. Proposition 4.5.
Let F be a non-commutative integrable system on a Pois-son manifold ( M, Π) , where M = T r × B s and F is projection on the secondcomponent. It is assumed that the Hamiltonian vector fields X f , . . . , X f r areindependent at every point of M . (1) If g ∈ C ∞ ( M ) , then g is Cas-basic if and only g is in involution withevery function which is constant on the fibers of F ; (2) Every pair of Cas-basic functions on M is in involution; (3) If g is Cas-basic, then its Hamiltonian vector field X g on M is ofthe form X F = P ri =1 ψ i X f i , where each ψ i is a Cas-basic functionon M .Proof. Suppose that g ∈ C ∞ ( M ) is in involution with every function whichis constant on all fibers of F . Then X f i [ g ] = { g, f i } = 0 for i = 1 , . . . , r ,hence g is F -basic, g = h ◦ F for some function h on B s . If k ∈ C ∞ ( B s ),then k ◦ F is constant on the fibers of F , so that { h, k } B ◦ F = { g, k ◦ F } = 0 , where we have used that F is a Poisson map. It follows that { h, k } B = 0 forall functions k on B s , hence that g (= h ◦ F ) is Cas-basic. This shows oneimplication of (1), the other one is clear. (2) is an easy consequence of (1).Consider now the Hamiltonian vector field X g of a Cas-basic function g on M . In view of (1), X g [ h ] = { h, g } = 0 for every function h which isconstant on the fibers of F , hence X g is tangent to the fibers of F . Sincethe fibers of F are spanned at every point by the Hamiltonian vector fields X f , . . . , X f r , there exist smooth functions ψ , . . . , ψ r on M , such that X g = r X i =1 ψ i X f i . The functions ψ i are F -basic, because X h [ ψ i ] = 0 for every function h whichis constant on the fibers of F . Indeed, for such a function h , we have that { g, h } = 0 and { f i , h } = 0 for i = 1 , . . . , r , so that0 = X { g,h } = [ X h , X g ] = r X i =1 ( X h [ ψ i ] X f i + ψ i [ X h , X f i ]) = r X i =1 X h [ ψ i ] X f i and the result follows from the independence of X f , . . . , X f r . (cid:3) Now, we can give a proposition that generalizes proposition 3.2 to thenon-commutative setting, which has formally the same shape up to the factthat r , formerly half of the rank of the Poisson structure Π, stands now for , AND POL VANHAECKE rank of the non-commutative integrable system, and up to the fact that thefunctions λ ji that appear below, are now proved to be Cas-basic, and notsimply F -basic. Proposition 4.6.
Let ( T r × B s , Π , F ) be an non-commutative integrable sys-tem of rank r , where F = ( f , . . . , f s ) is projection on the second component.There exists a ball B s ⊂ B s , also centered at , and there exist Cas-basicfunctions λ ji , such that the r vector fields Y i := P rj =1 λ ji X f j , ( i = 1 , . . . , r ) ,are the fundamental vector fields of a Hamiltonian torus action of T r on T r × B s . We can now turn our attention to the proof of proposition 4.6.
Proof.
As in Step 1 of the proof of proposition 3.6,we obtain the existenceof a family of R r -valued F -basic functions λ , . . . , λ r such that ˜Φ, definedas in (3.2), induces a T r -action on T r × B s , where B s is an s -dimensionalball, contained in B s . As in Step 2, we expand the fundamental vectorfields Y , . . . , Y r of the action in terms of the Hamiltonian vector fields X f , . . . , X f r , Y i = r X j =1 λ ji X f j . The proof that the vector fields Y i are Poisson vector fields requires an extraargument: we show that the relations X f i [ λ ki ] = 0 = X λ ji [ λ ki ] = 0 which wereused in (3.5) still hold, by showing that the functions λ ji are Cas-basic (recallthat Cas-basic functions are in involution). To do this, it suffices to showthat if a vector field on T r × B s of the form Z = P ri =1 ψ i X f i is periodicof period 1, then each of the coefficients ψ i is Cas-basic. Let Z be such avector field and consider Z := r X i =1 ψ i ( m ) X f i , where m is an arbitrary point in T r × B s . Then the restriction of Z to F − ( F ( m )) is periodic of period 1. Let h be an F -basic function on T r × B s ,and let us denote the (local) flow of X h by Φ t . Since[ X h , Z ] = r X i =1 ψ i ( m ) [ X h , X f i ] = 0 , for | t | sufficiently small, the flow of Z starting from Φ t ( m ) is also periodicof period 1. Since the coefficients of Z are the unique continuous functionssuch that Z = Z on F − ( F ( m )) and such that the flow of Z from everypoint has period 1, it follows that ψ i (Φ t ( m )) = ψ i ( m ) for | t | sufficientlysmall. Taking the limit t X h [ ψ i ] = 0 for every F -basicfunction on T r × B s . Thus, ψ i is Cas-basic, for i = 1 , . . . , r . so that the CTION-ANGLE COORDINATES 25 proof of Step 2 remains valid, amounting to the fact that the vector fields Y i on W are Poisson vector fields, L Y i Π = 0, which leads in view of (3.4) to r X j =1 X λ ji ∧ X f j = 0 . (4.1)We show that these vector fields are Hamiltonian, where the Hamiltonianscan be taken as a F -basic functions. The key point is that all coordinateswhich appear all along this step should now be taken with respect to coor-dinates adapted to f , . . . , f r . More precisely, we choose some m ∈ T r × B s in F − (0), and we construct in some neighborhood W ′ of m a system ofcoordinates ( f , g , . . . , f r , g r , z , . . . , z n − r )in which the Poisson structure takes the form given in equation (2.1). Ofcourse, the functions z , . . . , z n − r are F -basic again (and therefore dependon f , . . . , f s only), so that they can be defined in p − ( p ( W ′ )), an open subsetwhich we also call W ′ for the sake of simplicity. As before, we make no nota-tional distinction between the functions f , . . . , f r , z , . . . , z n − r , consideredas functions on p ( W ′ ) ⊂ B n − r , and the functions f , . . . , f r , z , . . . , z n − r themselves, defined on W ′ .In view of Proposition 4.5(3), in the previous system of coordinates, wehave that, since the functions λ ji are Cas-basic, X λ ji = r X k =1 (cid:0) ∂µ ji ∂f k ◦ F (cid:1) X f k , Hence, (4.1) gives X j We finally get to theaction-angle theorem, for standard Liouville tori of a non-commutative in-tegrable system. Theorem 4.7. Let ( M, Π) be a Poisson manifold of dimension n , equippedwith a non-commutative integrable system F = ( f , . . . , f s ) of rank r , andsuppose that F m is a standard Liouville torus, where m ∈ M F ,r ∩ U F . Thenthere exist R -valued smooth functions ( p , . . . , p r , z , . . . , z s − r ) and R / Z -valued smooth functions ( θ , . . . , θ r ) , defined in a neighborhood U of F m ,and functions such that (1) The functions ( θ , . . . , θ r , p , . . . , p r , z , . . . , z s − r ) define an isomor-phism U ≃ T r × B s ; (2) The Poisson structure can be written in terms of these coordinatesas Π = r X i =1 ∂∂θ i ∧ ∂∂p i + s − r X k,l =1 φ k,l ( z ) ∂∂z k ∧ ∂∂z l ;(3) The leaves of the surjective submersion F = ( f , . . . , f s ) are given bythe projection onto the second component T r × B s , in particular, thefunctions p , . . . , p r , z , . . . , z s − r depend on the functions f , . . . , f s only.The functions θ , . . . , θ r are called angle coordinates , the functions p , . . . , p r are called action coordinates and the remaining coordinates z , . . . , z s − r arecalled transverse coordinates .Proof. Conditions (1) and (2), in view of lemma 3.5, propositions 4.2 and 3.6imply that there exist on a neighborhood U ′ of F m in M on the one hand F -basic functions z , . . . , z s − r and on the other hand Cas-basic functions CTION-ANGLE COORDINATES 27 p , . . . , p r , such that p := ( p , . . . , p r , z , . . . , z s − r )and F define the same foliation on U ′ , and such that the Hamiltonian vectorfields X p , . . . , X p r are the fundamental vector fields of a T r -action on U ′ ,where each has period 1; the orbits of this torus action are the leaves ofthe latter foliation. In view of theorem 2.1, there exist on a neighborhood U ′′ ⊂ U ′ of m in M , R -valued functions θ , . . . , θ r such thatΠ = r X j =1 ∂∂θ j ∧ ∂∂p j + s − r X k,l =1 φ k,l ( z ) ∂∂z k ∧ ∂∂z l . The end of the proof goes along the same lines as the end of the proof oftheorem 3.8. (cid:3) Appendix: non-commutative integrability on Poissonmanifolds In the symplectic context, the terms superintegrability, non-commutativeintegrability, degenerate integrability, generalized Liouville integrability andMischenko-Fomenko integrability refer to the case when the Hamiltonianflow admits more independent constants of motions than half the dimensionof the symplectic manifold [3, 8, 9, 17, 16]. All these names correspond tonotions which are equivalent, at least locally. Similarly, the definition of anon-commutative integrable system on a Poisson manifold, which we havegiven in section 4 (definition 4.1) admits different locally equivalent formula-tions, which each have their own flavor. We illustrate this in this appendix,by giving an abstract geometrical formulation in terms of foliations, and aconcrete geometrical formulation in terms of Poisson maps.For both geometrical formulations, the notion of polarity in Poisson ge-ometry is a key element. Let m be an arbitrary point of a Poisson manifold( M, Π). The polar of a subspace Σ ⊂ T ∗ m M is the subspace Σ pol ⊂ T ∗ m M ,defined by Σ pol := { ξ ∈ T ∗ m M | Π m ( ξ, Σ) = 0 } . Notice that the polar of Σ pol can be strictly larger than Σ, because Π m mayhave a non-trivial kernel. When Σ = Σ pol , we say that Σ is a Lagrangiansubspace.Let F and G be two foliations on the same Poisson manifold ( M, Π).For m ∈ M we denote by T ⊥ m F the subspace of T ∗ m M , consisting of allcovectors which annihilate T m F , the tangent space to the leaf of F , passingthrough M . If F is defined around m by functions f , . . . , f s , then T ⊥ m F isspanned by d m f , . . . , d m f s . We say that F is polar to G if T ⊥ m F = ( T ⊥ m G ) pol ,for every m ∈ M ; also, F is said to be a Lagrangian foliation if F is polarto F . , AND POL VANHAECKE Definition 5.1. Let ( M, Π) be a Poisson manifold. An abstract non-commutative integrable system of rank r is given by a pair ( F , G ) of foliationson M , satisfying(1) F is of rank r and G is of corank r ;(2) F is contained in G (i.e., each leaf of F is contained in a leaf of G );(3) F is polar to G .This definition generalizes the definition of an abstract integrable systemon ( M, Π), which is simply a Lagrangian foliation F on M .In the following proposition we prove the precise relation between defini-tions 4.1 and 5.1. Proposition 5.2. Let ( M, Π) be a Poisson manifold. (1) If F = ( f , . . . , f s ) is a non-commutative integrable system of rank r on ( M, Π) , then on U F ∩ M F ,r the pair of foliations ( F , G ) , definedby F := fol( f , . . . , f s ) and G := fol( f , . . . , f r ) is an abstract non-commutative integrable system of rank r ; (2) Given ( F , G ) an abstract non-commutative integrable system of rank r on ( M, Π) , there exists for every m in M a neighborhood U in M , and functions F = ( f , . . . , f s ) on U , such that F is a non-commutative integrable system of rank r on U .Proof. (1) Recall from paragraph 4.1 that the open subsets U F and M F ,r of M are defined by U F := { m ∈ M | d m f ∧ d m f ∧ . . . ∧ d m f s = 0 } ,M F ,r := { m ∈ M | X f , X f , . . . , X f r are independent at m } . On U F ∩ M F ,r the functions f , . . . , f r define a foliation G of corank r ; sim-ilarly, the functions f , . . . , f s define a foliation F on it of rank r (since r + s = dim M ). Obviously, F is contained in G . The condition that { f i , f j } = 0 for all 1 i r and 1 j s , implies that the Hamil-tonian vector fields X f , . . . , X f r are tangent to F at every point. For all m ∈ U F ∩ M F ,r , the Hamiltonian vector fields X f , . . . , X f r are independentat m , hence they span T m F . It follows that( T ⊥ m G ) pol = { ξ ∈ T ∗ m M | Π m ( ξ, d m f i ) = 0 for i = 1 , . . . , r } = { ξ ∈ T ∗ m M | ξ ( X f i ( m )) = 0 for i = 1 , . . . , r } = T ⊥ m F . It follows that F is polar to G , hence ( F , G ) is an abstract non-commutativeintegrable system.(2) Let m be an arbitrary point of M . In a neighborhood U of m , thereexist smooth functions f , . . . , f s , such that the level sets of f , . . . , f s andof f , . . . , f r define foliations, which coincide with F and G on U . Since F is polar to G , the functions f , . . . , f r are in involution with the functions f , . . . , f s and the Hamiltonian vector fields of the functions f , . . . , f r are CTION-ANGLE COORDINATES 29 linearly independent at all points of U . It follows that F := ( f , . . . , f s ) is anon-commutative integrable system of rank r on U . (cid:3) When both foliations F and G are given by fibrations F : M → P and G : M → L respectively, we have a commutative diagram of submersivePoisson maps: ( M, Π) ( L, Π L )( P, Π P ) / / G (cid:15) (cid:15) F z z ttttttttttt (5.1)where Π L is the zero Poisson structure on L . Moreover, item (3) in defini-tion 5.1 amounts for every m ∈ M to the equality: F ∗ ( T ∗ F ( m ) P ) = (cid:0) G ∗ ( T ∗ G ( m ) L ) (cid:1) pol . (5.2)Conversely, it is clear that we have the following proposition: Proposition 5.3. Suppose that (5.1) is a commutative diagram of submer-sive Poisson maps, where L has dimension r and is equipped with the zeroPoisson structure Π L , and P has dimension dim M − r . If (5.2) holds forevery m ∈ M , then the pair of foliations ( F , G ) defined on M by F and G is an abstract non-commutative integrable system of rank r on ( M, Π) . References [1] M. Adler, P. van Moerbeke, and P. Vanhaecke. Algebraic integrability, Painlev´e geom-etry and Lie algebras , volume 47 of Ergebnisse der Mathematik und ihrer Grenzgebiete.3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics andRelated Areas. 3rd Series. A Series of Modern Surveys in Mathematics] . Springer-Verlag, Berlin, 2004.[2] V. I. Arnold. Mathematical methods of classical mechanics . Springer-Verlag, NewYork, 1978. Translated from the Russian by K. Vogtmann and A. Weinstein, GraduateTexts in Mathematics, 60.[3] A. V. Bolsinov and B. Jovanovi´c. 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A treatise on the analytical dynamics of particles and rigid bodies .Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1988.With an introduction to the problem of three bodies, Reprint of the 1937 edition,With a foreword by W. McCrea. Camille Laurent-Gengoux, Laboratoire de Math´ematiques et Applications,UMR 6086 CNRS, Universit´e de Poitiers, Boulevard Marie et Pierre CURIE,BP 30179, 86962 Futuroscope Chasseneuil Cedex, France E-mail address : [email protected] Eva Miranda, Departament de Matem`atiques, Universitat Aut`onoma de Bar-celona, E-08193 Bellaterra, Spain E-mail address : [email protected] Pol Vanhaecke, Laboratoire de Math´ematiques et Applications, UMR 6086du CNRS, Universit´e de Poitiers, Boulevard Marie et Pierre CURIE, BP30179, 86962 Futuroscope Chasseneuil Cedex, France E-mail address ::