## Abstract

This paper is a survey article on bi-Hamiltonian systems on the dual of the Lie algebra of vector fields on the circle. Here, we investigate the special case where one of the structures is the canonical Lie–Poisson structure and the second one is constant. These structures, called affine or modified Lie–Poisson structures, are involved in the integrability of certain Euler equations that arise as models for shallow water waves.

## 1. Introduction

In the last 40 years or so, the Korteweg–de Vries equation (KdV) (Korteweg & de Vries 1895) has received much attention in the mathematical physics literature. Some significant contributions were made in particular by Gardner, Green, Kruskal and Miura—see Praught & Smirnov (2005) for a complete bibliography and a historical review. It is through these studies that the *theory of solitons* as well as the *inverse scattering method* emerged.

One remarkable property of the KdV equation, highlighted at this occasion, is the existence of an infinite number of first integrals. The mechanism by which these conserved quantities were generated is at the origin of an algorithm called the *Lenard recursion scheme* or *bi-Hamiltonian formalism* (Magri 1978; Gel'fand & Dorfman 1979). This is representative of infinite-dimensional systems known as *formally integrable*, in reminiscence of finite-dimensional, classical integrable systems (in the sense of Liouville). Other examples of bi-Hamiltonian systems are the Camassa–Holm equation (Fokas & Fuchssteiner 1981; Camassa & Holm 1993; Constantin 1998; Constantin & McKean 1999; Gesztesy & Holden 2003) and the Burgers equation (Burgers 1948).

One common feature of all these systems is that they can be described as the geodesic flow of some right-invariant metric on the diffeomorphism group of the circle or on a central real extension of it, the Virasoro group. Each left (or right)-invariant metric on a Lie group induces, by a reduction process, a canonical flow on the *dual of its Lie algebra*. The corresponding evolution equation, known as the *Euler equation*, is Hamiltonian relatively to some canonical *Poisson structure*. It generalizes the Euler equation of the free motion of a rigid body.1 In a famous article (Arnold 1966), Arnold pointed out that this formalism could be applied to the group of volume-preserving diffeomorphisms to describe the motion of an ideal fluid.2 Thereafter, it became clear that many equations from mathematical physics could be interpreted in the same way.

Gel'fand & Dorfman (1981) showed that the KdV equation can be obtained as the geodesic equation, on the Virasoro group, of the right-invariant metric defined on the Lie algebra by the *L*^{2} inner product (see also Ovsienko & Khesin 1987). Misiolek (1998) has shown that the Camassa–Holm equation, which is a one-dimensional model for shallow water waves, can also be obtained as the geodesic flow on the Virasoro group for the *H*^{1} metric.

While both the KdV and Camassa–Holm equations have a geometric derivation and are models for the propagation of shallow water waves, the two equations have quite different structural properties. For example, while all smooth periodic initial data for the KdV equation develop into periodic waves that exist for all times (Tao 2002), smooth periodic initial data for the Camassa–Holm equation develop either into global solutions or into breaking waves—see the papers by Constantin (1997, 2000), Constantin & Escher (1998*a*,*b*, 2000) and McKean (2004).

In this paper, we study the case of right-invariant metrics on the diffeomorphism group of the circle, Diff(*S*^{1}). However, note that a similar theory is probable without the periodicity condition, in which case some weighted spaces express how close the diffeomorphisms of the line are to the identity (Constantin 2000).

Each right-invariant metric on Diff(*S*^{1}) is defined by an inner product **a** on the Lie algebra of the group, Vect(*S*^{1})=*C*^{∞}(*S*^{1}). If this inner product is *local*, it is given by the expressionwhere *A* is an invertible symmetric linear differential operator. To this inner product on Vect(*S*^{1}) corresponds a quadratic functional (the energy functional)on the (regular) dual Vect^{*}(*S*^{1}). Its corresponding Hamiltonian vector field *X*_{A} generates the Euler equation

Among the Euler equations of this kind, we have the well-known *inviscid Burgers* equationand the Camassa–Holm shallow water equation (Fokas & Fuchssteiner 1981; Camassa & Holm 1993)Indeed, the inviscid Burgers equation corresponds to *A*=*I* (*L*^{2} inner product), whereas the Camassa–Holm equation corresponds to *A*=1−*D*^{2} (*H*^{1} inner product)—see Constantin & Kolev (2002, 2003).

The Burgers, KdV and Camassa–Holm equations are precisely bi-Hamiltonian relative to some second *affine* (Souriau 1997) compatible Poisson structure3 (cf. McKean 1979; Constantin & McKean 1999; Lenells 2004). Since these equations are special cases of Euler equations induced by *H*^{k} metric, it is natural to ask whether, in general, these equations have similar properties for any value of *k*. Constantin & Kolev (2006) have shown that this *was not the case*. There are no affine structures on Vect^{*}(*S*^{1}) which makes the Eulerian vector field *X*_{k}, generated by the *H*^{k} metric, a bi-Hamiltonian system, unless *k*=0 (Burgers) or 1 (Camassa–Holm). One similar result for the Virasoro algebra was given by Constantin *et al.* (2006). Here, we investigate the problem of finding a modified Lie–Poisson structure for which the vector field *X*_{A} is bi-Hamiltonian. We show, in particular, that for an operator *A* with constant coefficients, this is possible only if *A*=*aI*+*bD*^{2}, where .

In §2, we recall the definition of Hamiltonian and bi-Hamiltonian manifolds and the basic materials on bi-Hamiltonian vector fields. Section 3 contains a description of Poisson structures on the dual of the Lie algebra of a Lie group. Section 4 is devoted to the study of bi-Hamiltonian Euler equations on Vect^{*}(*S*^{1}); the main results are stated and proved.

For the description of modified affine Poisson structures, we rely on Gelfand–Fuks cohomology. Since the handling of this cohomology theory is not obvious, we derive in appendix A an elementary ‘hands-on’ computation of the first two Gelfand–Fuks cohomological groups of Vect(*S*^{1}).

## 2. Hamiltonian and bi-Hamiltonian manifolds

In this section, we recall the definitions and the well-known results on finite-dimensional smooth Poisson manifolds.

### (a) Poisson manifolds

A *symplectic manifold* is a pair (*M*, *ω*), where *M* is a manifold and *ω* is a closed non-degenerate 2-form on *M*, i.e. d*ω*=0, and for each *m*∈*M*, *ω*_{m} is a non-degenerate skew-symmetric bilinear map on *T*_{m}*M*.

Since a symplectic form *ω* is non-degenerate, it induces an isomorphism(2.1)defined via *i*_{x}*ω*(*Y*)=*ω*(*X*, *Y*). For example, this allows the definition of the *symplectic gradient X*_{f} of a function *f* by the relation . The inverse of the isomorphism (2.1) defines a skew-symmetric bilinear form *P* on the cotangent space *T*^{*}*M*. This bilinear form *P* induces itself a bilinear mapping on *C*^{∞}(*M*), the space of smooth functions , given by(2.2)and called the *Poisson bracket* of the functions *f* and *g*.

The observation that a bracket like (2.2) could be introduced on *C*^{∞}(*M*) for a smooth manifold *M*, without the use of a symplectic form, leads to the general notion of a *Poisson structure* (Lichnerowicz 1977).

A *Poisson (or Hamiltonian*4*) structure* on a *C*^{∞} manifold *M* is a skew-symmetric bilinear mapping (*f*, *g*)↦{*f*, *g*} on the space *C*^{∞}(*M*), which satisfies the *Jacobi identity*(2.3)as well as the *Leibnitz identity*(2.4)

When the Poisson structure is induced by a symplectic structure *ω*, the *Leibnitz identity* is a direct consequence of equation (2.2), whereas the *Jacobi identity* (2.3) corresponds to the condition d*ω*=0 satisfied by the symplectic form *ω*. In the general case, the fact that the mapping *g*↦{*f*, *g*} satisfies equation (2.4) means that it is a *derivation* of *C*^{∞}(*M*).

Each derivation on *C*^{∞}(*M*) corresponds to a smooth vector field, i.e. to each *f*∈*C*^{∞}(*M*) is associated a vector field *X*_{f}:*M*→*TM*, called the *Hamiltonian vector field* of *f*, such that(2.5)where is the *Lie derivative* of *g* along *X*_{f}.

Jost (1964) pointed out that, just like a derivation on *C*^{∞}(*M*) corresponds to a vector field, a bilinear bracket {*f*, *g*} satisfying the Leibnitz rule (2.4) corresponds to a field of bivectors, i.e. there exists a *C*^{∞} tensor field *P*∈*Γ*(⋀^{2}*TM*), called the *Poisson bivector* of (*M*, {., .}), such that(2.6)for all *f*, *g*∈*C*^{∞}(*M*).

*A bivector field P*∈*Γ*(*⋀*^{2}*TM*) *is the* Poisson bivector *of a Poisson structure on M if, and only if, one of the following equivalent conditions holds*:

[

*P*,*P*]=0,*where*[ , ]*is the Schouten–Nijenhuis bracket*5,*the bracket*(*f*,*g*)=*P*(*d*_{f},*d*_{g})*satisfies the Jacobi identity*,*and*[

*X*_{f},*X*_{g}]=*X*_{{f,}_{g}}*for all f*,*g*∈*C*^{∞}(*M*).

By the definition of the Schouten–Nijenhuis bracket (Vaisman 1994), we havefor all *f*, *g*, *h*∈*C*^{∞}(*M*), where → indicates the sum over circular permutations of *f*, *g*, *h*. Hence, all these expressions vanish together. ▪

The notion of a Poisson manifold is more general than that of a symplectic manifold. Symplectic structures correspond to non-degenerate Poisson structure. In this case, the Poisson bracket satisfies the additional property that {*f*, *g*}=0 for all *g*∈*C*^{∞}(*M*) only if *f*∈*C*^{∞}(*M*) is a constant, whereas for Poisson manifolds such non-constant functions *f* might exist, in which case they are called *Casimir functions*. Such functions are constants of motion for all vector fields *X*_{g}, where *g*∈*C*^{∞}(*M*).

On a Poisson manifold (*M*, *P*), a vector field *X*:*M*→*TM* is said to be *Hamiltonian* if there exists a function *f* such that *X*=*X*_{f}. On a symplectic manifold (*M*, *ω*), the necessary condition for a vector field *X* to be Hamiltonian is thatA similar criterion exists for a Poisson manifold (*M*, *P*) (Vaisman 1994). The necessary condition for a vector field *X* to be *Hamiltonian* is

### (b) Integrability

An *integrable system* on a symplectic manifold *M* of dimension 2*n* is a set of *n* functionally independent6 *f*_{1}, …, *f*_{n} which are *in involution*, such thatA Hamiltonian vector field *X*_{H} is said to be *(completely) integrable* if the Hamiltonian function *H* belongs to an integrable system. In other words, *X*_{H} is integrable if there exists *n* first integrals7 of *X*_{H}, *f*_{1}=*H*, *f*_{2}, … ,*f*_{n} which commute together.

At any point *x* where the functions *f*_{1}, …, *f*_{n} are functionally independent, the Hamiltonian vector fields generate a *maximal isotropic* subspace *L*_{x} of *T*_{x}*M*. When *x* varies, the subspaces generate what one calls a *Lagrangian distribution*, i.e. a sub-bundle *L* of *TM* whose fibres are maximal isotropic subspaces. In our case, this distribution is integrable (in the sense of Frobenius). The leaves of *L* are defined by the equationsA Lagrangian distribution which is integrable (in the sense of Frobenius) is called a *real polarization* and is a key notion in *Geometric Quantization*.

In the study of dynamical systems, the importance of integrable Hamiltonian vector fields is emphasized by the *Arnold–Liouville theorem* (Arnold 1997), which asserts that each compact leaf is actually diffeomorphic to an *n*-dimensional toruson which the flow of *X*_{H} defines a linear quasi-periodic motion, i.e. in angular coordinates *φ*^{1}, …, *φ*^{n}where (*ω*^{1}, …, *ω*^{n}) is a constant vector.

In the case of a Poisson manifold, it can be confusing to define an integrable system. However, we can use the symplectic definition on each symplectic leaf of the Poisson manifold.

### (c) Bi-Hamiltonian manifolds

Two Poisson brackets { , }_{P} and { , }_{Q} are *compatible* if any linear combinationis also a Poisson bracket. A *bi-Hamiltonian manifold* (*M*, *P*, *Q*) is a manifold equipped with two Poisson structures *P* and *Q* which are compatible.

*Let P and Q be two Poisson structures on M*. *Then, P and Q are compatible if, and only if, one of the following equivalent conditions holds*:

[

*P*,*Q*]=0,*where*[ , ]*is the Schouten–Nijenhuis bracket*,→{{

*g*,*h*}_{P},*f*}*Q*+{{*g*,*h*}_{Q},*f*}_{P}=0,*where*→*is the sum over circular permutations of f*,*g*,*h*,*and**for all f*,*g*∈*C*^{∞}(*M*).

By definition of the Schouten–Nijenhuis bracket (Vaisman 1994), we havefor all *f*, *g*, *h*∈*C*^{∞}(*M*). Hence, all these expressions vanish together. ▪

### (d) Lenard recursion relations

On a bi-Hamiltonian manifold *M*, equipped with two compatible Poisson structures *P* and *Q*, we say that a vector field *X* is (formally) *integrable*8 or *bi-Hamiltonian* if it is Hamiltonian for both structures. The reason for this terminology is that for such a vector field, there exists under certain conditions a hierarchy of first integrals in involution that may lead in certain cases to complete integrability, in the sense of Liouville. A useful concept for obtaining such a hierarchy of first integrals is the so-called *Lenard scheme* (McKean 1993).

On a manifold *M* equipped with two Poisson structures *P* and *Q*, we say that a sequence of smooth functions satisfies the *Lenard recursion relation* if(2.7)for all .

*Let P and Q be Poisson structures on a manifold M and let* *be a sequence of smooth functions on M that satisfies the Lenard recursion relation*. *Then* *the functions*, *H*_{k}, *are pairwise in involution with respect to both the brackets P and Q*.

Using skew symmetry of *P* and *Q* and relation (2.7), we getfor all . From this we deduce, by induction on *p*, thatfor all . It is then an immediate consequence thatfor all . ▪

Note that in the proof of proposition 2.2, the compatibility of *P* and *Q* is not needed.

Suppose now that (*M*, *P*, *Q*) is a bi-Hamiltonian manifold and that at least one of the two Poisson brackets, say *Q*, is *invertible*. In this case, we can define a (1, 1)-tensor fieldwhich is called the *recursion operator* of the bi-Hamiltonian structure. Kosmann-Schwarzbach & Magri (1990, 1996) have shown that, as a consequence of the compatibility of *P* and *Q*, the *Nijenhuis torsion* of *R*, defined byvanishes. In this situation, the family of Hamiltonianssatisfies the Lenard recursion relation (2.7). Indeed, this results from the fact thatfor every vector field *X* and every (1, 1)-tensor field *T* on *M* and that the vanishing of the Nijenhuis torsion of *R* can be rewritten asfor all vector field *X*.

This construction has to be compared with *Lax isospectral equation* associated to an evolution equation(2.8)

*L*,

*B*), called a

*Lax pair*, whose coefficients are functions of

*u*and in such a way that when

*u*(

*t*) varies according to equation (2.8),

*L*(

*t*)=

*L*(

*u*(

*t*)) varies according toThis equation has been formulated by Lax (1968) in order to obtain a hierarchy of first integrals of the evolution equation as eigenvalues or traces of the operator

*L*. This analogy between

*R*and

*L*is not casual and has been studied by Kosmann-Schwarzbach & Magri (1996). Many evolution equations which admit a Lax pair also appear to be bi-Hamiltonian systems generated by a recursion operator

*R*=

*PQ*

^{−1}.

In practice, we may be confronted to the following problem. We start with an evolution equation represented by a vector field *X* on a manifold *M*. We find two compatible Poisson structures *P* and *Q* on *M* that make *X* a bi-Hamiltonian vector field; but both *P* and *Q* are *non-invertible*. In this case, it is however still possible to find a Lenard hierarchy if the following algorithm works.

*Step 1*. Let *H*_{1} be the Hamiltonian of *X* for the Poisson structure *P* and let *X*_{1}=*X*. The vector field *X*_{1} is Hamiltonian for the Poisson structure *Q* by assumption; this defines Hamiltonian function *H*_{2}. We define *X*_{2} to be the Hamiltonian vector field generated by *H*_{2} for the Poisson structure *P*.

*Step 2*. Inductively, having defined the Hamiltonian function *H*_{k} and letting *X*_{k} be the Hamiltonian vector field generated by *H*_{k} for the Poisson structure *P*, we check if *X*_{k} is Hamiltonian for the Poisson structure *Q*. If the answer is yes, then we define *H*_{k+1} to be the Hamiltonian of *X*_{k} for the Poisson structure *Q*.

## 3. Poisson structures on the dual of a Lie algebra

### (a) Lie–Poisson structure

The fundamental example of a non-symplectic Poisson structure is the *Lie*–*Poisson structure* on the dual of a Lie algebra .

On the dual space of a Lie algebra of a Lie group *G*, there is a Poisson structure defined by(3.1)for *m*∈ and *f*, *g*∈*C*^{∞}(), called the *canonical Lie–Poisson structure.*9

The canonical Lie–Poisson structure has the remarkable property to be *linear*, that is the bracket of the two linear functionals is itself a linear functional. Given a basis of , the components10 of the Poisson bivector *P* associated to equation (3.1) are(3.2)where are the *structure components* of the Lie algebra .

### (b) Modified Lie–Poisson structures

Under the general name of *modified Lie–Poisson structures*, we mean an affine11 perturbation of the canonical Lie–Poisson structure on . In other words, it is represented by a bivectorwhere *P* is the canonical Poisson bivector defined by equation (3.2) and *Q*=(*Q*_{ij}) is a constant bivector on . Such a *Q*∈⋀^{2}^{*} is itself a Poisson bivector. Indeed, the Schouten–Nijenhuis bracketsince *Q* is a constant tensor field on .

The fact that *P*+*Q* is a Poisson bivector, or equivalently that *Q* is compatible with the canonical Lie–Poisson structure, is expressed using proposition 2.2 by the condition(3.3)for all *u*, *v*, *w*∈.

### (c) Lie algebra cohomology

In this section, we deal with left-invariant forms but, of course, everything we say may be applied equally to right-invariant forms up to a sign in the definition of the coboundary operator. On a Lie group *G*, a left-invariant *p*-form *ω* is completely defined by its value at the unit element *e*, and hence by an element of ⋀^{p}^{*}. In other words, there is a natural isomorphism between the space of left-invariant *p*-forms on *G* and ⋀^{p}^{*}. Moreover, since the exterior differential d commutes with left translations, it induces a linear operator ∂:⋀^{p}^{*}→⋀^{p+1}^{*} defined by(3.4)where the hat means that the corresponding element should not appear in the list. *γ* is said to be a *cocycle* if ∂*γ*=0. It is a *coboundary* if it is of the form *γ*=∂*μ* for some cochain *μ* in the dimension *p*−1. Every coboundary is a cocycle, i.e. ∂∘∂=0.

For every *γ*∈⋀^{0}^{*}=, we have ∂*γ*=0. For *γ*∈⋀^{1}^{*}=^{*}, we havewhere *u*, *v*∈. For *γ*∈⋀^{2}^{*}, we havewhere *u*, *v*, *w*∈.

The kernel *Z*^{p}() of ∂:⋀^{p}()→⋀^{p+1}() is the space of *p-cocycles* and the range *B*^{p}() of ∂:⋀^{p−1}()→⋀^{p}() is the spaces of *p-coboundaries*. The quotient space is the *p*th *Lie algebra cohomology* or *Chevaley–Eilenberg cohomology group* of . Note that in general the Lie algebra cohomology is different from the de Rham cohomology . For example, , but .

Each 2-cocycle *γ* defines a modified Lie–Poisson structure on . The compatibility condition (3.3) can be recast as ∂*γ*=0. Note that the Hamiltonian vector field *X*_{f} of a function *f*∈*C*^{∞}() computed with respect to the Poisson structure defined by the 2-cocycle *γ* is(3.5)

A special case of modified Lie–Poisson structure is given by a 2-cocycle *γ* which is a coboundary. If *γ*=∂*m*_{0} for some *m*_{0}∈, the expressionlooks like if the Lie–Poisson bracket had been ‘frozen’ at a point *m*_{0}∈ and for this reason some authors call it a *freezing* structure.

## 4. Bi-Hamiltonian vector fields on Vect^{*}(*S*^{1})

### (a) The Lie algebra Vect(*S*^{1})

The group of smooth orientation-preserving diffeomorphisms of the circle *S*^{1} is endowed with a smooth manifold structure based on the *Fréchet space C*^{∞}(*S*^{1}). The composition and the inverse are both smooth maps ×→, respectively →, so that is a Lie group (Milnor 1984). Its Lie algebra is the space Vect(*S*^{1}) of smooth vector fields on *S*^{1}, which is itself isomorphic to the space *C*^{∞}(*S*^{1}) of periodic functions. The Lie bracket12 on is given by

*The Lie algebra* Vect(*S*^{1}) *is equal to its commutator algebra*, *i.e.*

Any real periodic function *u* can be written uniquely as the sumwhere *w* is a periodic function of total integral zero and *c* is a constant. To be of total integral zero is the necessary and sufficient condition for a periodic function *w* to have a periodic primitive *W*. Hence, we have [*W*, 1]=*w*. Moreover, since [sin, cos]=1, we have proved that every periodic function *u* can be written as the sum of two commutators. ▪

### (b) The regular dual Vect^{*}(*S*^{1})

Since the topological dual of the Fréchet space Vect(*S*^{1}) is too big and not tractable for our purpose, being isomorphic to the space of distributions on the circle, we restrict our attention in the following to the *regular dual* , the subspace of Vect(*S*^{1})^{*} defined by linear functionals of the formfor some function *m*∈*C*^{∞}(*S*^{1}). The regular dual is therefore isomorphic to *C*^{∞}(*S*^{1}) by means of the *L*^{2} inner product13With these definitions, the *coadjoint action*14 of the Lie algebra Vect(*S*^{1}) on the regular dual Vect^{*}(*S*^{1}) is given by

Let *F* be a smooth real-valued function on *C*^{∞}(*S*^{1}). Its *Fréchet* derivative d*F*(*m*) is a linear functional on *C*^{∞}(*S*^{1}). We say that *F* is a *regular function* if there exists a smooth map *δF*:*C*^{∞}(*S*^{1})→*C*^{∞}(*S*^{1}) such thatThat is, the Fréchet derivative d*F*(*m*) belongs to the regular dual and the mapping *m*↦δ*F*(*m*) is smooth. The map *δF* is a vector field on *C*^{∞}(*S*^{1}), called the *gradient* of *F* for the *L*^{2} metric. In other words, a regular function is a smooth function on *C*^{∞}(*S*^{1}), which has a smooth *L*^{2} gradient.

Typical examples of *regular functions* on the space *C*^{∞}(*S*^{1}) are *linear functionals*where *u*∈*C*^{∞}(*S*^{1}). In this case, *δF*(*m*)=*u*. Other examples are *nonlinear polynomial functionals*where *Q* is a polynomial in derivatives of *m* up to a certain order *r*. In this case, the gradient of *F* is just the *Eulerian derivative*Note that the smooth function *F*_{θ}:*C*^{∞}(*S*^{1})→ defined by *F*_{θ}(*m*)=*m*(*θ*) for some fixed *θ*∈^{1} is not regular since d*F*_{θ} is the Dirac measure at *θ*.

A smooth vector field *X* on is called a *gradient* if there exists a *regular function F* on such that *X*(*m*)=*δF*(*m*) for all *m*∈. Observe that if *F* is a smooth real-valued function on *C*^{∞}(*S*^{1}), then its second Fréchet derivative is symmetric (Hamilton 1982), i.e.

For a regular function, this property can be rewritten as(4.1)for all *m*,*M*,*N*∈*C*^{∞}(*S*^{1}), i.e. the linear operator *δF*′(*m*) is symmetric for the *L*^{2}-inner product on *C*^{∞}(*S*^{1}) for each *m*∈*C*^{∞}(*S*^{1}). Conversely, a smooth vector field *X* on , whose Fréchet derivative *X*′(*m*) is a symmetric linear operator, is the gradient of the function(4.2)This can be checked directly, using the symmetry of *X*′(*m*) and an integration by parts. We will resume this fact in lemma 4.2.

*On the Fréchet space C*^{∞}(*S*^{1}) *equipped with the (weak) L*^{2} *inner product, a necessary and sufficient condition for a smooth vector field X to be a* gradient *is that its Fréchet derivative X*′(*m*) *is a symmetric linear operator*.

### (c) Hamiltonian structures on Vect^{*}(*S*^{1})

To define a *Poisson bracket* on the space of *regular functions* on , we consider a one-parameter family of linear operators *P*_{m} (*m*∈*C*^{∞}(*S*^{1})) and set(4.3)The operators *P*_{m} must satisfy certain conditions in order for equation (4.3) to be a valid Poisson structure on the regular dual .

A family of linear operators *P*_{m} on defines a Poisson structure on if equation (4.3) satisfies the following conditions:

{

*F*,*G*} is regular if*F*and*G*are regular,{

*G*,*F*}=−{*F*,*G*}, and{{

*F*,*G*},*h*}+{{*G*,*H*},*F*}+{{*H*,*F*},*G*}=0.

Note that the second condition simply means that *P*_{m} is a skew-symmetric operator for each *m*.

The canonical Lie–Poisson structure on given byis represented by the one-parameter family of skew-symmetric operators(4.4)where *D*=∂_{x}. It can be checked that all the three required properties are satisfied. In particular, we have

The *Hamiltonian* of a *regular* function *F* for a Poisson structure defined by *P* is defined as the vector field

*A necessary condition for a smooth vector field X on* *to be Hamiltonian with respect to the Poisson structure defined by a constant linear operator Q is the symmetry of the operator X*′(*m*)*Q for each m*∈.

If *X* is Hamiltonian, we can find a regular function *F* such thatMoreover, since *Q* is a constant linear operator, we haveand therefore, we getwhich is a symmetric operator since *Q* is skew symmetric and *δF*′(*m*) is symmetric. ▪

### (d) Hamiltonian vector fields generated by right-invariant metrics

A right-invariant metric on the diffeomorphism group Diff(*S*^{1}) is uniquely defined by its restriction to the tangent space to the group at the unity, hence by a *non-degenerate continuous inner product* **a** on Vect(*S*^{1}). If this inner product **a** is *local*, then according to Peetre (1959), there exists a linear differential operator(4.5)where *a*_{j}∈*C*^{∞}(*S*^{1}) for *j*=0, …, *N*, such thatfor all *u*, *v*∈Vect(*S*^{1}). The condition for **a** to be non-degenerate is equivalent for *A* to be a *continuous linear isomorphism* of *C*^{∞}(*S*^{1}).

In the special case where *A* has *constant coefficients*, the *symmetry* is traduced by the fact that *A* contains only even derivatives and the *non-degeneracy* by the fact that the *symbol* of *A*,has no root in .

The right-invariant metric on Diff(*S*^{1}) induced by a continuous linear invertible operator *A* gives rise to an *Euler equation*15 on Vect(*S*^{1})^{*}(4.6)where *m*=*Au*. This equation is Hamiltonian with respect to the Lie–Poisson structure on Vect(*S*^{1})^{*} with Hamiltonian function on Vect(*S*^{1})^{*} given byThe corresponding Hamiltonian vector field *X*_{A} is given by

The family of operatorscorresponding respectively to the Sobolev *H*^{k} inner product, have been studied by Constantin & Kolev (2002, 2003). The *Riemannian exponential map* of the corresponding geodesic flow has been shown to be a local diffeomorphism, except for *k*=0. This latter case corresponds to the *L*^{2} metric on Diff(*S*^{1}) and happens to be *singular*.

A non-invertible inertia operator *A* may induce in some cases a weak Riemannian metric on a *homogenous space*. This is the way to interpret Hunter–Saxton and Harry Dym equations as Euler equations (see Khesin & Misiolek 2003).

Theorem 4.1 is a generalization of theorem 3.7 in Constantin & Kolev (2006).

*The only continuous linear invertible operators**with constant coefficients*, *whose corresponding Euler vector field X*_{A} *is bi-Hamiltonian relative to some modified Lie–Poisson structure*, *are**where* *satisfy* . *The second Hamiltonian structure is induced by the operator**where D*=d/d*x and the Hamiltonian function is**where m*=*Au*.

We insist on the fact that the proof we give applies for an operator with *constant coefficients*. It would be interesting to study the case of a continuous linear invertible operator whose coefficients are *not constant*. Is there such an operator *A* with bi-Hamiltonian Euler vector field *X*_{A} relative to some modified Lie–Poisson structure? In this case, for which modified Lie–Poisson structures *Q* is there an Euler vector field *X*_{A} which is bi-Hamiltonian relatively to *Q*?

The proof is essentially the same as the one given by Constantin & Kolev (2006). A direct computation shows thatwhereandwhere .

Each modified Lie–Poisson structure on Vect^{*}(*S*^{1}) is given by a *local 2-cocycle* of Vect(*S*^{1}). According to proposition A.2 (see appendix A), such a cocycle is represented by a differential operator(4.7)where *m*_{0}∈*C*^{∞}(*S*^{1}) and . We will now show that there is no such cocycle for which *X*_{A} is Hamiltonian if the order ofis strictly greater than 2.

By virtue of proposition 4.1, a necessary condition for *X*_{A} to be Hamiltonian with respect to the cocycle represented by *Q* is thatis a symmetric operator. We haveand in particular, for *m*=1,Hence,whereasTherefore, letting , we getand this operator vanishes if and only if(4.8)But is the sum of two linear differential operators,which is of the order of less than 2*N*+2, andwhich is of the order of 4*N*+1 unless , which must be the case if equation (4.8) holds. Therefore, *m*_{0} has to be a constant. Let . Thenbecause *D* and *A* commute. The symmetry of the operator means(4.9)for all *m*,*M*,*N*∈*C*^{∞}(*S*^{1}). Since this last expression is trilinear in the variables *m*, *M* and *N*, the equality can be checked for complex periodic functions *m*, *M and N*. Let *m*=*Au*, *u*=e^{−ipx}, *M*=e^{−iqx} and *N*=e^{−irx} with *p*, *q*, *r*∈. We havewhereas

Now, we set *p*=*n*, *q*=−2*n*, *r*=*n* and must have(4.10)if *K*(*m*) is symmetric.

If *β*≠0, the leading term in the left-hand side of equation (4.10) is 24(−1)^{N}*a*_{2N}*βn*^{2N+4}, whereas the leading term of the right-hand side is 6(−1)^{N}2^{2N}*βn*^{2N+4}. Hence, unless *N*=1, we must have *β*=0.

On the other hand, if *β*=0, we must have *αs*_{A}(*n*)=α*s*_{A}(2*n*) for all . Thus, *α*=0 unless *N*=0. This completes the proof. ▪

### (e) Hierarchy of first integrals

In view of theorem 4.1, the next step is to find a hierarchy of first integrals in involution for the vector field *X*_{A}, whereand satisfy . The vector fieldis bi-Hamiltonian. It can be written aswhereand , or aswhere

The problem we get when we try to apply the Lenard scheme to obtain a hierarchy of conserved integrals is that both Poisson operators *P*_{m} and *Q* are non-invertible. However, *Q* is composed of two commuting operators, *A* which is invertible and *D* which is not. The image of *D* is the codimension 1 subspace, , of smooth periodic functions with zero integral. The restriction of *D* to this subspace is invertible with inverse *D*^{−1}, the linear operator which associates to a smooth function with zero integral its unique primitive with zero integral. Following Lax (1976), we are able to prove the following result.

*There exists a sequence* *of functionals*, *whose gradients G*_{k} *are polynomial expressions of u*=*A*^{−1}*m and its derivatives*, *which satisfies the Lenard recursion scheme*

It is worth noting that contrary to the result given by Lax (1976), for the KdV equation, the operators *G*_{k} are polynomials in *u*=*A*^{−1}*m* and not in *m*. In particular, there are non-local operators16, if *A*≠*aI*, for some .

Before giving a sketch of proof of this theorem, let us illustrate the explicit computation of the first Hamiltonians of the hierarchy. We start withWe define *X*_{1} to be the Hamiltonian vector field of *H*_{1} for the Lie–Poisson structure *P*_{m,}*X*_{1}(*m*) is in the image of *D* for all *m* and we can definewhich is the gradient of the second Hamiltonian of the hierarchyWe then compute *X*_{2}, the Hamiltonian vector field of *H*_{2} for *P*_{m},where . *X*_{2}(*m*) is in the image of *D* for all *m* and we can definewhich is the gradient of the third Hamiltonian of the hierarchy

So far, we obtain in this way a hierarchy of Hamiltonians satisfying the Lenard recursion relations for the Euler equation associated to the operator *A*.

(Burgers Hierarchy). For *A*=*I*, we obtain explicitly the whole *Burgers hierarchy*

(Camassa–Holm Hierarchy) For *A*=*I*−*D*^{2}, we obtain the *Camassa*–*Holm hierarchy*. The first members of the family areThe next integrals of the hierarchy are much harder to compute explicitly. One may consider Lenells (2005) and Loubet (2005) for further studies on the subject.

The proof is divided into two steps. We refer to Lax (1976) for the details.

*Step 1.* We show by induction that there exists a sequence of vector fields *G*_{k}, which is a polynomial expression of *u*=*A*^{−1}*m* and its derivatives and satisfies(4.11)

*Step 2*. We show that *G*_{k} is, for all *k,* the gradient of a function *H*_{k}. ▪

To prove Step 1, we suppose that *G*_{1}, …, *G*_{n} have been constructed satisfying equation (4.11) and we use lemmas 4.3 and 4.4,17 to show that *G*_{n+1} exists.

*Suppose that Q is a polynomial in derivatives of u up to order r such that**for all u*∈*C*^{∞}(*S*^{1}). *Then, there exists a polynomial G in derivatives of u up to order r*−1 *such that Q*=*DG*.

*We have**for all* .

To prove Step 2, it is enough to show that is a symmetric operator for all *k*, by virtue of lemma 4.2. We suppose that G_{1}, …, *G*_{n} are gradients and show first the following result.

*The operator**is symmetric for all m*∈*C*^{∞}(*S*^{1}).

We conclude then, like in Lax (1976), that itself is symmetric. We will give here the details of the proof of lemma 4.5, since the proof of the corresponding result for KdV in Lax (1976) is just a direct, hand-waving computation and does not apply in our more general case.

First, we differentiate the recurrence formula (4.11) and we obtain(4.12)and(4.13)Multiplying equation (4.12) by *Q* on the right, (4.13) by *P* on the right, and subtracting equation (4.13) from (4.12), we getUsing the fact thatwe finally getUsing the fact that *Q* satisfies the cocycle conditionwhich can be rewritten aswe getBut this last expression is zero becauseand . ▪

In the special case where the cocycle *γ* is a coboundary, i.e. when the second structure is a *freezing structure*, the algorithm used to generate a hierarchy of first integrals is known as the *translation argument principle* (Arnold & Khesin 1998; Khesin & Misiolek 2003). Let *H*_{λ} be a function on , which is a Casimir function of the Poisson structureThat is, for every function *F* one hasSuppose that *H*_{λ} can be expressed as a seriesThen, one can check that *H*_{0} is a Casimir function of and that for all *k*, the Hamiltonian vector field of *H*_{k+1} with respect to coincides with the Hamiltonian vector field of *H*_{k} with respect to . Furthermore, all the Hamiltonians *H*_{k} are in involution with respect to both the Poisson structures and the corresponding Hamiltonian vector fields commute with each other. In practice, to obtain such a Casimir function *H*_{λ}, one chooses a Casimir function *H* of the Poisson structure and then *translates the argument*

## Acknowledgments

This paper was written during the author's visit to the Mittag-Leffler Institute in October 2005, in conjunction with the Program on Wave Motion. The author wishes to extend his thanks to the Institute for its generous sponsorship of the program, as well as to the organizers for their work. The author also expresses his gratitude to David Sattinger for several remarks that helped to improve this paper.

## Footnotes

One contribution of 13 to a Theme Issue ‘Water waves’.

↵In this case, the group is just the rotation group,

*SO*(3).↵However, this formalism seems to have been extended to hydrodynamics before Arnold by Moreau (1959).

↵The affine structure on the Virasoro algebra, which makes the KdV equation a bi-Hamiltonian system, seems to have been first discovered by Gardner (1971) and for this reason, some authors call it the

*Gardner bracket,*see also Faddeev & Zakharov (1971).↵The expression ‘Hamiltonian manifold’ is often used for the generalization of Poisson structure in the case of infinite-dimension manifolds.

↵The Schouten–Nijenhuis bracket is an extension of the Lie bracket of vector fields to skew-symmetric multivector fields (see Vaisman 1994).

↵This means that the corresponding Hamiltonian vector fields

*X*_{f1}, …,*X*_{fn}are independent on an open dense subset of*M*.↵A first integral is a function which is constant on the trajectories of the vector field.

↵This terminology is used for the evolution equations in infinite dimension.

↵Here, d

_{m}*f*, the differential of a function*f*∈*C*^{∞}() at*m*∈, is to be understood as an element of the Lie algebra .↵In what follows, the convention for lower or upper indices may be confusing since we shall deal with tensors on both and . Therefore, we emphasize that the convention we use in this paper is the following: upper indices correspond to contravariant tensors on and therefore covariant tensors on , whereas lower indices correspond to covariant tensors on and therefore contravariant tensors on .

↵A Poisson structure on a linear space is

*affine*if the bracket of two linear functionals is an affine functional.↵This corresponds to the opposite of the usual Lie bracket of vector fields.

↵In the sequel, we use the notation

*u*,*v*, … for elements of and*m*,*n*, … for elements of to distinguish them, although they all belong to*C*^{∞}(*S*^{1}).↵The coadjoint action of a Lie algebra on its dual is defined aswhere

*u*,*v*∈,*m*∈, and the pairing is the standard one between and .↵The second order geodesic equation corresponding to a one-sided invariant metric on a Lie group can always be reduced to a first-order quadratic equation on the dual of the Lie algebra of the group: the Euler equation (see Arnold & Khesin (1998) or Kolev (2004)). The generality of this reduction was first revealed by Poincaré (1901) and applied to hydrodynamics by Arnold (1966).

↵Note that our

*m*corresponds to*u*in the notations of Lax (1976).↵The proof of lemma 4.3 can be found in Olver (1993), while the proof of lemma 4.4 can be found in Lax (1976).

↵Using a theorem of Peetre (1959), a local cochain can be characterized by the condition

↵Recall that ∂

*m*is the linear differential operator defined by- © 2007 The Royal Society