Generically, Arnold-Liouville Systems Cannot be Bi-Hamiltonian

We state and prove that a certain class of smooth functions said to be BH-separable is a meagre subset for the Fr\'echet topology. Because these functions are the only admissible Hamiltonians for Arnold-Liouville systems admitting a bi-Hamiltonian structure, we get that, generically, Arnold-Liouville systems cannot be bi-Hamiltonian. At the end of the paper, we determine, both as a concrete representation of our general result and as an illustrative list, which polynomial Hamiltonians $H$ of the form $H(x,y)=xy+ax^3+bx^2y+cxy^2+dy^3$ are BH-separable.


Introduction
In the late 1970s and early 1980s appeared the notion of bi-Hamiltonian system in a seminal paper by F. Magri [18]. This theory was widely developped a few years later by F. Magri and C. Morosi in [19]. At the same time, another fundamental work was done in this field by I. Gel'fand and I. Dorfman [10] but, in fact, this theory had already its roots in the socalled "Lenard recursion formula" as well explained in [21]. In the late 1980s and early 1990s some works were carried out to well understand the link between, on the one hand, those of the Hamiltonian systems which are completely integrable in the sense of Arnold-Liouville and, on the other hand, the bi-Hamiltonian systems [4,5,9,15]. We emphasize the fact that the studies in the first three references above presented a completely different approach to the problem than the fourth. Here we focus only on the first approach which is the original approach presented by Magri and Morosi. We also point out that the present work only considers the compatibility of symplectic forms and never examines the compatibility between degenerate Poisson structures. Moreover, other approaches have been made to provide interesting geometric structures on the phase space of an integrable Hamiltonian system. We can quote for example works of the early 2000's on separability [8,13] or a very recent work on Haantjes structures [22]. The work presented in this article is a natural questioning of the results stated in [4,5,9] leading us to examine a very particular class of Hamiltonians verifying a separability property which seems quite restrictive. Roughly speaking, such Hamiltonians need to be locally separable through a change of local coordinates which also separates the initial (action) variables of the Hamiltonian. A function satisfying this property will be said BH-separable and we will denote by S BH their set. In a recent work [3] we studied the topology of a class of smooth functions that we called locally separable, constituting a set denoted by S. Precisely, we defined these latter functions in the following way: a function 1 H : (x 1 , . . . , x n ) → H(x 1 , . . . , x n ) defined on an open ball centered at the origin O is called a locally separable function if it can be locally separated in the sense that there exists a smooth diffeomorphism ϕ (depending on H), fixing O, from an open neighborhood V of O onto its image W ϕ : V → W, (x 1 , . . . , x n ) → (u 1 (x 1 , . . . , x n ), . . . , u n (x 1 , . . . , x n )), such that for all (x 1 , . . . , x n ) ∈ V we have H(x 1 , . . . , x n ) = H 1 (u 1 (x 1 , . . . , x n )) + · · · + H n (u n (x 1 , . . . , x n )).
The work previously mentioned explained how such separable functions are generic among smooth functions. Now, in the case of the BH-separability studied in the present work the condition imposed on the functions is more restrictive and consists of asking that not only the function H belongs to S but also that one of the smooth diffeomorphisms ϕ separating H also separates the coordinates x i in the sense that for all i ∈ 1, n , we could write for some functions x ij , ∀i ∈ 1, n , x ij (u j ).
So, finally H is BH-separable if we can find new local coordinates (u 1 , . . . , u n ) around O such that ∀i ∈ 1, n , ∀(k, l) ∈ 1, n 2 , k = l, The aim of this paper is to prove that, this time, unlike the case of locally separable functions studied in [3], the set of BH-separable functions is meagre. 2 So, it will conclude that, generically, Arnold-Liouville completely integrable Hamiltonian systems cannot be bi-Hamiltonian or, in other words, that generically the complete integrability in the sense of Arnold-Liouville cannot be explained and got by the existence of an underlying bi-Hamiltonian structure.
In the first part we will recall the origin of this special separability and how it appears in the context of the bi-Hamiltonian systems. In the second part we will give our main result, stated in two different frames, namely:  In the third and last part, we will determine among a special class of polynomial Hamiltonians which of them are BH-separable.

The mechanics origin of BH-separability
In order to understand why we are interested in this strange property of BH-separability we have to recall briefly how it appeared in the frame of bi-Hamiltonian systems, and hence the genesis of the problem.
Let (M, ω, H) be a Hamiltonian system, namely a symplectic manifold (with a dimension denoted by 2n) and a smooth function (the Hamiltonian) H on M . This Hamiltonian system is said to be completely integrable if there are n functions f 1 , . . . , f n satisfying the following conditions [1,2,16]: (1) they are first integrals of the Hamiltonian vector field X H , i.e., where { , } denotes the Poisson bracket associated with the symplectic form ω; (2) they Poisson-commute, or are in involution, in the sense that (3) they are functionaly independent, at least on a dense open subset of M , i.e., df 1 ∧ · · · ∧ df n = 0.
In this situation, let us denote by F the function . . , f n (m)).
Because of the above property (3), F is a submersion. 3 If c is in the range of F then F −1 (c) is a submanifold of M . If F is a proper map, then F is a fibration (Ehresmann theorem [7]). If we assume that its fibres are connected then they are n-tori T n . Moreover, any of these fibres F −1 (c) possesses a tubular neighborhood Ω which can be identified, up to a symplectomorphism Φ, to the symplectic manifold (U × T n , ω 0 ), where U is some open set of R n (that we can assume, without loss of generality, to be some open ball centered at the origin) endowed with coordinates (x 1 , . . . , x n ) and ω 0 denotes the canonical symplectic form defined as . . , θ n being angle coordinates on the torus; this symplectic identification between Ω and U × T n can be furthermore factorized via a map ϕ between the basis of the fibration F and the basis U of the trivial fibration pr 1 : U × T n → U , in such a way that the following diagram be commutative Finally, the Hamiltonian H in the coordinates (x i , θ i ) depends only on the x i , so is a basic function with respect to the fibration. All these properties constitute the statement of the classic Arnold-Liouville theorem [1,2]; the coordinates (x 1 , . . . , x n , θ 1 , . . . , θ n ) defined by the map Φ are called action-angle coordinates. In what follows we will call Arnold-Liouville system the semilocal model described by the Arnold-Liouville theorem for a completely integrable Hamiltonian system verifying the conditions of the theorem, namely (U × T n , ω 0 , H), where H is a basic function for the trivial fibration, so a function depending only on the action coordinates x i . In the early eighties the concept of bi-Hamiltonian system was introduced by Magri and Morosi [19]. Roughly speaking, the main idea of this theory is the following: if there exists a second symplectic form, compatible in a natural sense with the first one, and if the initial Hamiltonian field is also Hamiltonian with respect to this new symplectic structure, then it would be possible to generate mechanically first integrals in involution and so get the complete integrability of the Hamiltonian vector field. Precisely, two symplectic forms ω 0 and ω 1 on a manifold M define a (1, 1) tensor field J, called in this context a recursion operator, by the formula This torsion appears also in the frame of Kähler manifolds and its vanishing is an integrability condition of the eigenspaces distribution. Now a vector field X is said bi-Hamiltonian with respect to such compatible symplectic forms if it is a Hamiltonian vector field for both forms. Actually, we generally assume that i X ω 0 = −dH (so X is Hamiltonian with respect to ω 0 ) and that L X ω 1 = 0 (ω 1 is invariant by X), condition which can also be written di X ω 1 = 0, meaning that X is only locally Hamiltonian with respect to ω 1 . What was proved by Magri and Morosi in [19] is that, for such a bi-Hamiltonian vector field X, the functions Tr J k , k ∈ N constitute a Poisson commuting family of first integrals of X. It produces the result that, in the case where at each point m of the manifold, J m owns the maximum possible number of eigenvalues, namely n, the spectrum {λ 1 , . . . , λ n } of J provides also a Poisson commuting family of first integrals of X and so that X is completely integrable as soon as dλ 1 ∧ · · · ∧ dλ n = 0. So, roughly speaking one can say that a bi-Hamiltonian vector field is completely integrable, at least in the case where its recursion operator possesses a convenient spectrum. So the natural question is: what about the converse? If X is some completely integrable Hamiltonian system, is it possible to find a bi-Hamiltonian structure which explains this integrability, in the sense that the eigenvalues of the associated recursion operator constitute a Poisson commuting family of first integrals for the given field? At the end of the eighties and the beginning of the nineties some works [4,5,9] were done to answer this question for an Arnold-Liouville system, i.e., they study the existence of such a bi-Hamiltonian structure on the whole manifold U × T n and not only locally, this latter problem being almost obvious. The answer was negative because such an existence implies very restrictive conditions on the Hamiltonian, namely that it must be BH-separable in the sense that we have defined it above in this article. Precisely the next result was proved [4,5,9]: the θ i denoting the angle coordinates on the tori T n and H is a function (the Hamiltonian) of the x i . Let us assume that: (i) This system is bi-Hamiltonian, i.e., there exists on M 0 a second symplectic form ω 1 , compatible with ω 0 , such that the Hamiltonian (with respect to ω 0 ) vector field X H is also Hamiltonian with respect to ω 1 (at least locally).
(ii) The number of eigenvalues of the recursion operator associated to the couple (ω 0 , ω 1 ) is maximum (so equal to n).

(iii) The Hamiltonian H is non-degenerate in the sense that the Hessian matrix
Then there are around each point of U local coordinates (u 1 , . . . , u n ) such that In particular the Hamiltonian H is BH-separable.
At that time, some examples were provided [4,5,9] showing some Arnold-Liouville systems for which no such bi-Hamiltonian structure exists, because their Hamiltonians were not BHseparable. The aim of this paper is to state and prove that not only there exists Arnold-Liouville systems with no bi-Hamiltonian structure but also that these cases are not exceptions but the general rule.
3 The main theorem: from function space to integrability

Frame and preliminary results
In what follows we will work on the space C ∞ (U, R) of real functions defined on an open set U of R n that we can assume to be the open ball B(O, 1). This space will be endowed with its structure of Fréchet metric space [12].
The first statement allows us to only consider functions for which the 1-jet at O is zero.
where the a i are real constants so under a change of coordinates (x 1 , . . . , x n ) → (u 1 , . . . , u n ) which separates the x i , we get , that for all (k, l) ∈ 1, n 2 , with k = l, the vanishing of the quantities So we can always suppose, without loss of generality, that all our functions have O as a critical point, i.e., that their 1-jet at O is zero. From now on we will suppose that we are in this situation.
In [4,5], or [9], some examples of polynomial Hamiltonians which are not BH-separable were given. For example, as we will see in the last section, a Hamiltonian H as H(x, y) = xy + xy 2 is not BH-separable. However, because its Hessian is not degenerate, this function is locally separable in the sense defined in [3] and mentioned in the introduction; this is a direct consequence of Morse's lemma. So, in a general way, we have Remark 3.3. Of course the obvious inclusion of S BH into S givesS BH ⊂S. So, if we use a recent work [3] which studiesS we can have some informations aboutS BH Nevertheless, it would be impossible by this mean to conclude that it is a meager subset of the space of smooth functions.
Finally, we recall the Faà Di Bruno's formula [6,14] which will furnish a precious tool in the proof of our main theorem in the next subsection: If H and φ are functions of class C k then for all where for any smooth function f of n variables f (k) denotes its differential of order k, P k denotes the set of partitions of 1, k and |P | denotes the cardinality of P .
Formulae of Faà Di Bruno type, but for partial derivatives were given by M. Hardy [11] and T-W. Ma [17]. But, for our purpose, it is not necessary to use them.

Main theorem (function space topology version)
Now, we can state and prove the first version of the main result of this paper, purely stated in terms of function space topology. Proof . First, let us introduce some useful notations. If f is a smooth function, let us denote by T k (f ) the homogeneous part of degree k of its Taylor's expansion at the origin; so T k (f ) belongs to the space H n k of the homogeneous polynomials of degree k with n indeterminates X 1 , . . . , X n . We will denote also by J k O (f ) its k-jet at the origin defined as which is nothing but its Taylor polynomial at O and with degree k. Now, by the very definition of the set S BH , the fact that the function f belongs to S BH means that there exists: • a change of variables U : (x 1 , . . . , x n ) → (u 1 , . . . , u n ), • n functions g 1 , . . . , g n of class C ∞ depending on only one variable, 4 This Faà Di Bruno's formula is somewhat frightening at first sight! Perhaps a little illustration might clarify this. Let us consider the simple case where H depends only on one variable and let us try to visualise what P is, what B is and so on. For the choice k = 3, we get where actually all the terms in H, or its derivatives, in the right hand side, have to be understood as H • n 2 functions (f ij ) 1≤i,j≤n of class C ∞ depending only on one variable, such that, in a neighborhood of the origin, the functions f, x 1 , . . . , x n can be written as f (x 1 , . . . , x n ) = g 1 (u 1 ) + · · · + g n (u n ), ∀i ∈ 1, n , Let us denote respectively by G and F the functions G(u 1 , . . . , u n ) = g 1 (u 1 ) + · · · + g n (u n ) and By means of these functions we can write the relations (3.1) in a condensed way, namely where W is an open neighborhood of the origin in R n . Differentiating the relation F Using Jacobian matrix, this relation can be written where Com(M ) denotes the comatrix of a matrix M . It follows from this formula that each partial derivative ∂u i ∂xp is a rational fraction P ip Q ip of the numbers f rs (0). The coefficients of this rational fraction are independent from U , F and G; they only depend on the indices i and p. On the other hand, So, let us introduce the application where H n 1 denotes the space of homogeneous polynomial functions of degree 1 with n indeterminates. Then the previous discussion shows that if f belongs to S BH , then T 1 (f ) belongs to the range of the application L 1 .
Next, using the Faà di Bruno's formula, we get the differential of order 2 at O of U , namely We deduce from this relation that each of the second partial derivatives ∂ 2 u i ∂xp∂xq (0) is a rational fraction R ipq S ipq of the numbers f rs (0) and f rs (0), the latter not appearing in the denominators. Here again the coefficients of these fractions depend only on the indices i, p and q and not on the functions U , F and G.
On the other hand, we have also So, let us introduce the map where L 2 (a 1 , . . . , a n , b 1 , . . . , b n , A, M ) := As in the case of first derivatives, if f belongs to S BH , then T 2 (f ) belongs to the range of the application L 2 . What happens for higher degrees of derivation? Using induction, we can easily prove that, around O, for all positive integer k, each of k-th order partial derivatives of the u i are rational fractions of all the derivatives of the functions f rs . It is true for k = 1. Indeed, we have seen that at the origin but actually it is also true near O because F (O) is invertible and so F (u) remains invertible for u close to O. Now, if we have where R is some rational function with m variables t 1 , . . . , t m , then where, for the sake of simplicity, we denote by t j in ∂t j ∂x i k+1 something which is some f We can write this last formula by ordering terms with respect to the length of P , namely in such a way that if a function f belongs to S BH then for all positive integers k, T k (f ) is in the range of L k . The dimension of the domain of the map L k is k n + n 2 whereas those of its codomain H n k is which we can also write as showing that it is a polynomial function of k with a degree n − 1. So if n ≥ 3, its degree is a least quadratic and so its increase at infinity ensures that for k sufficiently large it will be strictly larger than the linear expression k n + n 2 and so the inequality holds. On the contrary, for n = 2, the two compared dimensions are respectively 6k and k + 1 and so the inequality 6k < k + 1 is always false. For that reason we will have to deal separately with the cases n ≥ 3 and n = 2. Let us begin with the case n ≥ 3. Let k be an integer such that the inequality (3.2) holds. Then the range of L k has a Lebesgue measure equal to zero. We know that R kn × R kn 2 is a countable union of compact sets (K i ) i≥0 , so the range of L k is a countable union of the compact sets (L k (K i )) i≥0 and each L k (K i ) has no interior points since its measure is zero. Because T k is an open and continuous map we get that each T −1 k (L k (K i )) is a closed set with no interior points. So we deduce that S BH is meagre.
It remains to examine the case n = 2. In this case, we can proceed as previously but by replacing T k (f ) by the k-jet of f , namely J k O (f ). This time we introduce a map where R k [x, y] denotes the space of homogeneous polynomial functions with two indeterminates and total degree ≤ k. With these notations the expression . For k > 9 we have 6k + 1 < (k+1)(k+2) 2 , so the range of L k has its measure equal to zero. Thus, we can apply the same arguments as in the case n ≥ 3 and conclude that S BH is a meagre set.

Main theorem (integrability version): Arnold-Liouville vs BH-systems
where x denotes here the column vector of the x i . This function f can be arbitrarily close to H according as to the choice made on A and does not belong to D k , so H ∈D k . So D is a meagre set as a countable union of nowhere dense closed sets.
According to Theorems 2.1 and 3.5 we can state that: Theorem 3.7. Generically, an Arnold-Liouville system admits none bi-Hamiltonian structure in the sense of Theorem 2.1.
By this statement we understand the following result: the set H BH of Hamiltonians H ∈ C ∞ (B (O, 1), R) for which the Arnold-Liouville system (M 0 , ω 0 , H) admits a bi-Hamiltonian structure (with the conditions of Theorem 2.1) is meagre for the Fréchet's topology.
Proof . Using the notation of the previous lemma we can write According to Theorem 2.1, we have the inclusion (N D ∩ H BH ) ⊂ S BH , so Theorem 3.5 implies that N D ∩H BH is meagre. But D ∩H BH ⊂ D so, using the previous lemma, we get that D ∩H BH is meagre. It results in that H BH is meagre as the union of two meagre sets. 4 BH-separability of a special class of polynomial Hamiltonians

Preliminary considerations
The previous sections have shown that among the smooth functions, the BH-separable ones are rather "rare". Nevertheless, of course, they do exist. In this last part, we will determine among a class of particular Hamiltonians with two variables, 5 which are BH-separable. This study will be doubly useful: first, obviously, to obtain these rare candidates but also to visualize perfectly and concretely the previously studied phenomenon of scarcity in a restricted family of functions.
A challenging study would be to determine (in the case of two variables) which Hamiltonians of the form H = Q + C, where Q is a non-degenerate quadratic form and C a general cubic polynomial, are BH-separable. Let us begin with a few remarks.
• The quadratic part Q has a signature (2, 0), (1, 1) or (0, 2) because it is assumed to be non-degenerate. So, using a linear change of coordinates it can be written x 2 + y 2 , x 2 − y 2 or −x 2 − y 2 . Unfortunately, under a change of coordinates, even under a linear change of coordinates, the property of BH-separability does not necessarily remain. Indeed, the problem is that the coordinate change must separate both functions x and y. So, the only study of these three cases would not allow us to cover all the situations.
• The study of the three cases Q = x 2 + y 2 , Q = x 2 − y 2 , Q = −x 2 − y 2 (the third would be essentially the same as the first) is tedious and leads to awful calculations, even using some symbolic computing tool. For all those reasons we will present here only one of these cases and we have chosen the case where Q has signature (1, 1). Moreover, we will deal with Q = xy instead of Q = x 2 − y 2 , the first case being a little more pleasant than the last. So we will search, inside the family xy + ax 3 + bx 2 y + cxy 2 + dy 3 indexed by (a, b, c, d) ∈ R 4 , the BH-separable elements. We can remark that this family is a 4-dimensional affine subspace of the space of smooth functions.
We can also remark that all these Hamiltonians are locally separable in the sense studied in [3], because they are Morse's functions at the origin. 6 • If H is BH-separable there is some change of variable (x, y) → (u, v) around the origin O, fixing O, and which separates H, x and y. In other words we have six smooth functions f , g, h, k, H 1 , H 2 , each of them depending only on one variable (u or v) such that and Replacing if necessary f by f − f (0) we can assume that f (0) = 0. Since the change of variable fixes O, then g(0) = 0. In the same way we can suppose that h(0) = k(0) = 0. Because the considered map is a change of variable, then necessarily (f (0), g (0)) = (0, 0) and (h (0), k (0)) = (0, 0). Because in these families of Hamiltonians the 1-jet at (0, 0) is zero, it remains zero in coordinates u, v and the quadratic part (the Hessian) is tensorial, so its signature is invariant under any change of coordinates, in particular in coordinates u, v. So, writing and comparing the two expressions of H in coordinates u, v given by ( * ) we get where ( a 1 a 2 a 3 a 4 ) denotes the Jacobian matrix at the origin of the map (u, v) → (x, y). Because the signature of the considered quadratic form is (1, 1), then necessarily α = 0 and β = 0 and because α = a 1 a 3 and β = a 2 a 4 , none of the real numbers a 1 , a 2 , a 3 , a 4 is zero. So, without loss of generality, we can assume that x = u + v (or y = u + v). Indeed, because a 1 = f (0) = 0, the map u → U = f (u) is a local diffeomorphism around 0 fixing 0 and we can write u = f −1 (U ); in the same way, because a 2 = g (0) = 0, we get locally v = g −1 (V ) and so we can write x = U + V and y = h f −1 (U ) + k g −1 (V ) . The same thing could be done with the functions h and k instead of f and g in order to get y = U + V if we would wish it. 7
These two ODE are equivalent by the change k → −k. Rewriting the first one we get Let us suppose that A = 0 and let us denote L = Ap + Bu + 1. Then the last differential equation can be written as 7 If we chose to deal with Q = x 2 − y 2 , or with Q = x 2 + y 2 the situation would be quite different. Indeed, we get (a1u + a2v) 2 ± (a3u + a4v) 2 = αu 2 + βv 2 , a condition which does not imply that the four coefficients a1, a2, a3, a4 are not zero. For example a2 may be zero but, in this case, a1 = 0 and a4 = 0 since a1a4 − a2a3 = 0. It follows that in this case we can always assume that x = u + q(v) and y = p(u) + v or x = p(u) + v and y = u + q(v).
Taking into account that L(0) = 1, we deduce that L 2 2 + AC − B 2 u 2 2 − (B + Ak)u − 1 2 = 0 and so that L 2 = − AC − B 2 u 2 + 2(B + Ak)u + 1, or L = − AC − B 2 u 2 + 2(B + Ak)u + 1, in order to finally get We get the formula for q replacing k by −k: where A = 2c + 6kd, B = 2(b + kc) and C = 6a + 2kb. So, the previous formulae (4.3) and (4.4) give the necessary form of a change of variables (u, v) → (x = u + v, y = p(u) + q(v)) which could separate the Hamiltonian H in turn. Let us denote the functions p and q being replaced by the expressions obtained in (4.3) and (4.4) and H u k v l the derivatives ∂ k+l H ∂u k ∂v l . All the terms H u k v l (0, 0) have to be zero. Using some symbolic computing tool 8 we verify that But higher derivatives do not vanish. Indeed we get successively: H u 2 v 2 (0, 0) = −24 cdk 4 + (3ad − bc)k 2 + ab , (4.5) and This last equality is especially interesting and we will base the next discussion on it by distinguishing several cases. We emphasize that all of the following calculations were performed using a symbolic calculation tool.
First case: c = 0 Let us first notice that in this case we must assume d = 0 because if d = 0, expressions of the functions p and q have no sense (denominators vanish). So the case c = d = 0 will have to be studied separately.
In this case, the condition (4.7) is, of course, satisfied and the two others ((4.5) and (4.6)) give respectively a 3dk 2 + b = 0 and 3dk 3 − bk + 3a −3bdk 2 − 9adk + b 2 = 0. (4.8) These two conditions invite us to look at the special case a = 0. If a = 0 then the first of the previous relations is satisfied and the second one gives, because k = 0, b 3dk 2 − b 2 = 0.
So H is BH-separable. Now, if a = 0, necessarily b = −3dk 2 . In this case the second part of the relation (4.8) writes 27kd 2dk 3 + a 2dk 3 − a = 0. Because k and d are not zero, we get two possibilities: a = ±2dk 3 . Replacing these values in relation (4.6) we do not get zero so the case a = 0 cannot lead to a case of BH-separability when c = 0.
This condition leads us to consider the two subcases d = 0 and 3a − ck 2 = 0.
Replacing them in H we get that the term in u 2 v 2 is equal to −6ab, so necessarily a = 0 or b = 0. For example, if a = 0, we get that the term in u 2 v 3 is equal to 2kb 3 and so vanishes if and only if b = 0. If we suppose b = 0, we get in the same manner that a = 0. So, once we have c = d = 0, right away we get a = b = 0 and so H = xy, which is a trivial case already contained in the first family for d = 0. The last thing that we have to consider is that we have prioritized the variable x over the variable y and so we have to exchange x and y to get forgotten Hamiltonians, namely H = xy + dx 3 , d ∈ R, H = xy + dx 3 + 3cx 2 y + 3dk 2 xy 2 + ck 2 y 3 , (c, d, k) ∈ R 2 × R * . (4.10) The two forms (4.9) and (4.10) are not equivalent because, for example, H = xy + y 3 belongs to one of the families (4.9) and not to one of the families (4.10). We can summarize all this discussion by this result: