Canonoid and Poissonoid Transformations, Symmetries and BiHamiltonian Structures

We give a characterization of linear canonoid transformations on symplectic manifolds and we use it to generate biHamiltonian structures for some mechanical systems. Utilizing this characterization we also study the behavior of the harmonic oscillator under canonoid transformations. We present a description of canonoid transformations due to E.T. Whittaker, and we show that it leads, in a natural way, to the modern, coordinate-independent definition of canonoid transformations. We also generalize canonoid transformations to Poisson manifolds by introducing Poissonoid transformations. We give examples of such transformations for Euler's equations of the rigid body (on $\mathcal{so}^\ast (3) $ and $ so^\ast (4)$) and for an integrable case of Kirchhoff's equations for the motion of a rigid body immersed in an ideal fluid. We study the relationship between biHamiltonian structures and Poissonoid transformations for these examples. We analyze the link between Poissonoid transformations, constants of motion, and symmetries.

1. Introduction. BiHamiltonian systems are, in a nutshell, dynamical systems described by a vector field that is Hamiltonian with respect to two distinct Poisson (or symplectic) structures and two associated (possibly distinct) Hamiltonian functions. Under certain additional hypothesis, possessing a biHamiltonian structure is enough to guarantee the integrability of the system (see for example [17]).
During the last few decades it has been shown that many integrable systems are in fact biHamiltonian, consequently, biHamiltonian structures are now an important paradigm for understanding integrability.
In some cases, a new Hamiltonian structure can be obtained with a transformation of coordinates. This may be possible when, on a symplectic manifold, the transformation changes the Hamiltonian characterization of a Hamiltonian vector field. Such transformations (in the case the symplectic manifold is R 2n and the symplectic form the standard one) were dubbed "canonoid" and popularized by Saletan and Cromer [26], and by Currie and Saletan [7], but they were know well before the 1970s, in fact, they were already present in the 1904 edition of the classical book of Whittaker [30].
This type of transformations include the well known canonical ones. The main difference between these transformations is that, while the canonoid ones are specific to the problem considered, the canonical ones preserve the Hamiltonian form of every Hamiltonian system on the manifold, and leave invariant the symplectic structure. Therefore, canonical transformations cannot be used to generate different symplectic structures. Strictly canonoid transformations (i.e., those canonoid transformations that are not canonical), in contrast, change the symplectic structure and only preserve the Hamiltonian form of some chosen Hamiltonian systems, hence, they can be used to generate different symplectic structures.
Canonoid transformations are the subject of about 20 papers, among them, we cite here the ones more closely related to the content of our article. A first set of papers concerns primarily the characterization of canonoid transformations and their relations with canonical transformations: [7,23,4,29,5,3]. A second set deals primarily with applications of canonoid transformations to the analysis of Hamiltonian systems: [27,14].
In this paper we use a modern geometrical definition of canonoid transformation based on locally Hamiltonian vector fields. This definition coincides to the so called quasi-canonical transformations of Marmo [19] and reduces to the definition of Saletan and Cromer [26] in the simplest case of a topologically trivial system, or at least when considering only local expressions for the system. By generalizing the approach of [10], we obtain simple explicit conditions for linear canonoid transformations on R n . We use this method to analyze some examples, including the harmonic oscillator in R 4 . We also recall the approach of Whittaker [30] and show that the modern definition of canonoid transformation we employ follows naturally from such approach.
Moreover, we extend this type of transformations to the case of Poisson manifolds, by introducing a generalization of the canonoid transformations that we dub Poissonoid transformations. This type of transformations, as far as we know, have not been studied before, and they allow us to find biHamiltonian structures in the case of Poisson manifolds. The Casimirs of the new Poisson structures found this way provide first integrals of the Hamiltonian system. Furthermore, if the Poisson structures are compatible, the integrability of the systems follows from the theory of biHamiltonian systems.
We also study the relationship between Linear Poissonoid transformations and biHamiltonian structures in some examples, namely Euler's equations for the rigid body (on so * (3) and so * (4)) and an integrable case of Kirchhoff's equations for the motion of a rigid body immersed in an ideal fluid.
We conclude with a study of the relations among infinitesimal Poissonoid transformations, Noether theorem and master symmetries, generalizing to the Poisson case some results obtained in [3] for canonoid transformations.
Our aim is to provide to the non-specialists an introduction to canonoid transformation and biHamiltonian systems through the analysis of several examples. For the specialists, we highlight the new definition of Poissonoid transformations, the role played by simple linear canonoid (and Poissonoid) transformations in the determination of several biHamiltonian structures, and the relationship between Poissonoid transformations, integrals of motion, and symmetries.
The paper is organized as follows. In section 2 we recall some essential facts concerning Poisson geometry, symplectic geometry and biHamiltonian structures, and we set the notations employed in the rest of the article. Section 2 can be skipped by readers already familiar with these topics. In section 3 we introduce canonoid transformations on symplectic manifolds and we study examples of linear canonoid transformations. In section 4 we analyze how the superintegrable structure of some simple systems behaves under linear canonoid transformations. In section 5 we translate into more modern language the characterization of canonoid transformations given in Whittaker [30]. In section 6, we extend the idea of canonoid transformations to Poisson manifolds by introducing Poissonoid transformations, and we give several examples of such transformations. In the last section we analyze the link between infinitesimal Poissonoid transformations, master symmetries and constants of motion. The definition of Hamiltonian system can be generalized to systems that are Hamiltonian in a neighborhood of each point of the manifold (M, π). Here, by a neighborhood of a point x we mean an open set containing x. Moreover, we use a definition of locally Hamiltonian given in [15]. Note that this definition differs from the one used in [16] and [6]. Definition 2.5. A vector field X on a Poisson manifold (M, π) is locally Hamiltonian if for every x ∈ M there is a neighborhood U of x and a smooth function H U defined on this neighborhood such that X = π · dH U , that is X is Hamiltonian in U with the locally defined Hamiltonian H U . A triplet (M, π, X ) as above is called a locally Hamiltonian system.
Let (M, π) be a Poisson manifold of dimension d. In a neighborhood U of a point p ∈ M , with local coordinates x = (x 1 , . . . , x d ), the bivector field π can be written as and the vector field We recall that a function F ∈ C ∞ (M ) such that {F, H} = 0 is called a constant of motion or first integral of the Hamiltonian system. Definition 2.6. Let (M, π) be a Poisson manifold. Let (M, π, H) be a Hamiltonian system on M with Hamiltonian vector field X H . A vector field ξ on M is an infinitesimal symmetry of X H if L ξ X H = 0. Moreover, ξ is called a Poisson infinitesimal symmetry of (M, π, H) if it is an infinitesimal symmetry of both π and H, that is if L ξ π = 0 and L ξ H = ξ[H] = 0.
A Poisson infinitesimal symmetry ξ is a (locally) Hamiltonian infinitesimal symmetry of (M, π, H) if, in addition, ξ is (locally) Hamiltonian. That is, there is a (locally defined) function K such that ξ = π · dK.
Note that, in the non-degenerate (symplectic) case, Poisson infinitesimal symmetries coincide with locally Hamiltonian infinitesimal symmetries.
It is important to keep in mind there is a distinction between Poisson infinitesimal symmetries of (M, π, H) and infinitesimal symmetries of the X H = π · dH. The following proposition clarifies the relationship between the two types of symmetries Proposition 2. If ξ is a Poisson infinitesimal symmetry of the Hamiltonian system (M, π, H), then it is also an infinitesimal symmetry of X H .
Proof. Let α be an arbitrary 1-form. Since X H = π · dH, we have (1) If ξ is a Poisson infinitesimal symmetry of (M, π, H), then L ξ π = 0 and L ξ H = 0. Since α is arbitrary, by the equation above we have L ξ X H = 0.
The converse of this proposition is clearly not true in general, even when π is non-degenerate [3].
Note that Hamiltonian infinitesimal symmetries are very important because, through Noether's theorem (see below), they generate constants of motion, which are very useful in the process of reduction the Hamiltonian system. Theorem 2.7 (Noether's theorem). Let (M, π, H) be a Hamiltonian system. If F is a constant of motion then its vector field is a Hamiltonian infinitesimal symmetry. Conversely, each Hamiltonian infinitesimal symmetry is the Hamiltonian vector field of a constant of motion, which is unique up to a (time dependent) Casimir function.
Proof. Suppose F is a constant of motion. Then X F = π · dF is a Hamiltonian vector field, so that L X F π = 0. Moreover, since X H [F ] = 0 , we have that Thus X F is a Hamiltonian infinitesimal symmetry. Now suppose that V is a Hamiltonian infinitesimal symmetry of (M, π, H). Since V is Hamiltonian there is a function F such that V = X F = π · dF . Since it is an infinitesimal symmetry of (M, π, H) we have that and F is a constant of motion. IfF is another function that satisfies XF = V = X F , then 0 = XF − X F = π · (d(F − F )), 6 GIOVANNI RASTELLI AND MANUELE SANTOPRETE soF −F must be a Casimir of π, since if we apply the above formula to an arbitrary function G we have that Definition 2.8. A vector field X on a Poisson manifold (M, π) is called a Poisson vector field if L X π = 0.
In particular, it follows that any locally Hamiltonian vector field is Poisson: If X is a locally Hamiltonian vector field on (M, π), then it is a Poisson vector field.
The converse is not true in general. For example, if the Poisson structure is trivial, then any vector field is Poisson, while the only Hamiltonian vector field is the trivial one. In the special case of a symplectic manifold, a vector field is Poisson if and only if it is locally Hamiltonian (see Proposition 5). Theorem 2.9 (Weinstein's splitting theorem ). Let (M, π) be a Poisson manifold, let x ∈ M be an arbitrary point and denote the rank of π at x by 2r. There exists a coordinate neighborhood U of x with coordinates (q 1 , . . . q r , p 1 , . . . , p r , z 1 , . . . z s ) centered at x, such that, on U , where the functions φ kl are smooth functions which depend on z = (z 1 , . . . , z s ) only, and which vanish when z = 0. Such local coordinates are called splitting coordinates, centered at x. In particular, if there is a neighborhood V of x such that the rank is constant and equal to 2r, then there exists a coordinate neighborhood U of x with coordinates (q 1 , . . . q r , p 1 , . . . , p r , z 1 , . . . z s ) centered at x, such that, on U , Moreover, π is locally of the form above, in any splitting coordinates on M . Such coordinates are called Darboux coordinates.
Remark 1. For a given point x on a Poisson manifold M , splitting coordinates are not unique. The Poisson structure, which is defined in a neighborhood of z = 0 by the second term of (2), however, is unique up to a Poisson diffeomorphism (see for example [15]).
Proposition 4. Let (M, π) be a Poisson manifold and let x ∈ M . Suppose X is a locally Hamiltonian vector field on M . Let U be a neighborhood of x with splitting coordinates (q, p, z) = (q 1 , . . . , q r , p 1 , . . . , p r , z 1 , . . . , z s ), and H U such that X = π · dH U = π(·, dH U ). In these coordinates Thus (q(t), p(t), z(t)) is an integral curve of X if and only if Hamilton's equations hold: . . , r, and k = 1, . . . , s, where the functionsφ ij (z) depend on the choice of splitting coordinates. Moreover, if the rank is locally constant at the point x, then the vector field, written in Darboux coordinates, is and Hamilton's equations take the simpler forṁ for i = 1, . . . , r, and k = 1, . . . , s.

Symplectic structures.
We give a brief account of symplectic structures, for more details see [1,20]. From the above definition it follows that, associated to ω, there is a Poisson bivector π such that a symplectic structure is a regular Poisson structure of maximal rank. The basic link between the Poisson bivector π and the symplectic form ω is that they are associated to the same Poisson bracket {F, H} = π(dF, dH) = ω(X F , X H ), that is dF, π · dH = dF, X H . On the other hand, by definition, ω(X H , v) = dH · v, and so ω · X H , v = dH, v , whence since ω = (ω ) −1 (see [1]). Thus π · dH = ω · dH, for all H, and π = ω .

GIOVANNI RASTELLI AND MANUELE SANTOPRETE
Definition 2.13. A vector field on a symplectic manifold (M, ω) is locally Hamiltonian if for every x ∈ M there is a neighborhood U of x and a smooth function H U defined on this neighborhood such that i X ω = dH U , that is X is Hamiltonian in U with the locally defined Hamiltonian H U .
If the manifold M has zero first group of real homology H 1 (M, R), then all local Hamiltonian vector fields are globally Hamiltonian [11].

Remark 2.
Another equivalent way to define locally Hamiltonian vector fields is the following. A vector field is locally Hamiltonian if there exists a closed 1-form α such that X = ω · α. In fact, if X = ω · α, then i X ω = α. Therefore, saying that α is closed is equivalent to saying that i X ω is closed.
Theorem 2.14 (Darboux' Theorem for the symplectic case). Let (M, ω) be a symplectic manifold of dimension 2n, then for each point x ∈ M there exists a neighborhood U of x with coordinates (q 1 , . . . , q n , p 1 , . . . , p n ) (called canonical coordinates ) such that Proof. See [1]. Proposition 6. Let (M, ω) be a symplectic manifold and let x ∈ M . Let X be a locally Hamiltonian vector field on M . Let U be a neighborhood of x with canonical coordinates (q, p) = (q 1 , . . . , q n , p 1 , . . . , p n ), and letH U be a function such that i X ω = dH U . In these coordinates where 1 and 0 define the n × n identity and the zero matrix, respectively.
Thus (q(t), p(t)) is an integral curve of X if and only if Hamilton's equations hold:q Proof. See [1].

BiHamiltonian structures.
Here we briefly introduce some of the most important facts about biHamiltonian structures (see also [17,18] for details).
We now recall the local coordinate representations of the pull-back of two forms and bivectors, and of the push-forward of vector fields, useful in some of the computations of the following sections.
Let M and N be manifolds, f : M → N be a smooth map, and let ρ be a two-form on N . By f * ρ we denote the pull-back of ρ by f , that in local coordinates takes the form Now suppose f is a diffeomorphism, and let π be a bivector field on N , then the pull-back of π by f in local coordinates is If X is a vector field on M , then the pushforward f * X of X by f , in local coordinates, takes the form Proof. Since the bracket has the following property (see [28]): it follows that [π, π] = 0 implies [f * π, f * π] = 0. Corollary 1. Let M be a manifold and let (N, π 1 , π 2 ) be a biHamiltonian manifold, that is π 1 and π 2 are compatible. Suppose f : M → N is a diffeomorphism. Then f * π 1 and f * π 2 are compatible and (M, f * π 1 , f * π 2 ) is a biHamiltonian manifold.
we have Comparing the two equations above, by Proposition 7 we obtain that f * [π 1 , Corresponding to the bivector fields π 1 and π 2 we can define the Poisson brackets {F, G} 1 = π 1 (dF, dG) and {F, G} 2 = π 2 (dF, dG). With these notations we give the following Definition 2.16. Let (M, π 1 , π 2 ) be a biHamiltonian manifold and suppose there exist functions H 1 and H 2 on M for which for every function F on M . Then X is called a biHamiltonian vector field.
The importance of biHamiltonian structures lies in the fact that, in certain situations, they can be used to show complete integrability. We do not give a complete account, but the main idea is that one can use them to build a set of first integrals in involution by constructing a biHamiltonian hierarchy [17,15,2] Definition 2.17. Let (M, π 1 , π 2 ) be a biHamiltonian manifold. A biHamiltonian hierarchy on M is a sequence of functions {F i } i∈Z such that The following lemma explains why a biHamiltonian hierarchy yields functions in involution. Proof.
so that {F i , F j } 1 = 0 by skew-symmetry. Hence, the F i 's are in involution with respect to the Poisson bracket {·, ·} 1 , and also with respect to {·, 3. Canonoid Transformations. Definition 3.1. Let (M, ω) be a symplectic manifold, and let X be a locally Hamiltonian vector field on M , that is, for each x ∈ M , there is a neighborhood U of x and locally defined function This is equivalent to saying that, for each y ∈ M , there is a neighborhood V of y and a locally defined function K V such that i X (f * ω) = f * dK V . This is also equivalent to L X (f * ω) = 0.
Remark 3. By Proposition 6 the previous definition means that, in a neighborhood U of each point x ∈ M the system of equations associated with i X ω = dH U can be written in Darboux coordinates as: and the system associated with i f * X ω = dK V can be written, in Darboux coordinates on the neighborhood V of f (x), as: so that the transformation f carries the system of Hamilton's equations (5) again into a system of Hamilton's equations (6).
We modify an example found in [10] to give a general framework to construct linear canonoid transformations. Let us consider Hamiltonian systems on the symplectic manifold (M, ω) = (R 2n , ω) and let x = (q, p) be Darboux coordinates on R 2n , then the symplectic form can be written as ω = dq i ∧dp i . In this case, a dif- Any quadratic Hamiltonian can be written as where S is a real symmetric constant 2n × 2n matrix. With these notations, the Hamiltonian vector field corresponding to H is JSx, and Hamilton's equations take the formẋ = JSx.
Hamilton's equations above define a linear Hamiltonian system with constant coefficients. Consider the transformation f : R 2n → R 2n , defined by X = f (x) = Ax, with A an invertible matrix. Then the vector field X is transformed to f * X = AJSA −1 X, and the system of Hamilton's equations is transformed into a new system of 2n differential equationsẊ = AJSA −1 X, expressed in terms of the variables X = (Q, P). In general, the new system does not have the canonical structure, that is, it is not necessarily true that there exists a Hamiltonian K(X) such thaṫ and thus not every transformation of this type is canonoid. However, it is easy to see that, in order to preserve the canonical structure, we must have for some symmetric matrix C. We can rewrite this condition as A t JAJS = −A t CA.
It follows that the existence of a symmetric matrix C is equivalent to the symmetry condition with Γ = A t JA = −Γ t . Thus, in this case, the condition for having a canonoid transformation reduces to equation (7).

Remark 4.
If the transformation is canonical, then the matrix A is symplectic (A T JA = J), and thus, Γ = J and the condition is satisfied. The same is true if Γ = aJ ( with a = 0).

Remark 5. If
A represents a rescaling of the given coordinates, namely, then, ΓJ = JΓ = B, a diagonal matrix determined by The transformation is canonoid if and only if (7) holds, i.e.

BS = SB.
When the rescaling is a point-transformation, then a i b i = 1, and the transformation is canonical.
We now find more explicit conditions to have canonoid transformations. Write the matrices Γ and S in terms of n × n blocks as follows: where λ t = −λ, ν t = −ν, α t = α, and γ t = γ. The equations ΓJS = SJΓ lead to the system Proof. Showing that Γ = aJ, for some constant a = 0, satisfies the conditions is trivial. Conversely, consider the particular case α = γ = 0. We find that µ must commute with every n × n matrix, and therefore µ = a1. Choosing α = β = 0 we find λ = 0. From β = γ = 0 it follows that ν = 0. Hence Γ = aJ, and in addition, from A t JA = aJ, it follows that JAJ = −a(A −1 ) t . We finally find that C = a(A −1 ) t SA −1 and the new Hamiltonian is K(X) = 1 2 X t CX. If A is symplectic it holds that K(X) = H(x). If a = 1, then A is not symplectic and we find K(X) = aH(x).
We now show that, given a canonoid transformation, it is possible to find an additional symplectic structure and an additional first integral. Consequently, canonoid transformations can be used to find biHamiltonian structures and to study the integrability of Hamiltonian systems.
Let ω 1 = ω be the symplectic form defined by Then, the Hamiltonian vector field with Hamiltonian H satisfies the equation for all v ∈ R 2n . Similarly, let Ω be the symplectic form in the "transformed space" defined by: Then, the Hamiltonian vector field X K with Hamiltonian K satisfies the equation We can use f to define a new canonical form by pulling back the canonical form Ω: which gives an explicit expression of the symplectic form ω 2 in terms of the matrix A.
Then, by pulling back the equation Ω(X K , v) = dK · v, we can write Hence, the vector field X H is Hamiltonian with respect to the symplectic form ω 1 and with respect to the symplectic form ω 2 .
Example 3.2. Let S be such that β = α = 0, and γ = 1. Then the equations reduce to Hence, from the first (or second) equation, λ = 0, and from the last equation µ t = µ. Then, by equation ( 8), the only requirements are that c t a = a t c (i.e. c t a is symmetric), and that 14 GIOVANNI RASTELLI AND MANUELE SANTOPRETE Example 3.3. We specialize the previous example. Let H = 1 2 (p 2 1 + p 2 2 ) then S is a 4 × 4 matrix with β = α = 0 and γ = 1. Suppose that a = 1, b = c = 0, and that Moreover, c t a = 0 and so it is symmetric. Therefore, this transformation satisfies the conditions obtained in the previous example. Then we can compute C as C = −JAJSA −1 . We obtain and hence K = 1 2 (mP 2 1 + nP 2 2 + 2lP 1 P 2 ). We compute ω 2 and H 2 as in (9) for this example. The transformation is given by the matrix where 1 and 0 are the 2 × 2 identity matrix and the zero matrix, respectively. The matrix d is given by Then the matrix representative of ω 2 is given by Note that the matrix A is symplectic if and only if l = 0 and m = n = 1, so that this transformation is symplectic if and only if it is the identity. The new Hamiltonian, obtained after some computations, is Clearly H 2 is a first integral of the system with Hamiltonian H, since {H, H 2 } = 0.
Example 3.4. We now consider a more interesting example, namely, the harmonic oscillator. In this case β = 0, and α = γ = 1. The conditions for having a canonoid transformation reduce to ν = λ , and µ = µ t (i.e. µ is symmetric). Suppose S is a 2 × 2 matrix, we consider the following subcases: where E 3 = Γ is the matrix of the new symplectic form. Moreover, so that K = 1 2 X t CX is the Hamiltonian of the transformed system. Transforming K back to the old coordinates yields the Hamiltonian (b) By putting b = c = 0 and taking a and d to be symmetric matrices, we have and µ = a t d is a symmetric matrix. Then If, in particular, we set a 11 = d 12 = a 22 = 0, a 12 = 1, d 11 = 1, and d 22 = 1, then we obtain W 2 = (q 1 q 2 + p 1 p 2 ), which is also a first integral, and the corresponding symplectic form has the following matrix representation If, instead, we set a 12 = d 12 = 0, a 11 = a 22 = d 11 = 1, and d 22 = −1 then we obtain the first integral and the corresponding symplectic form has the following matrix representation Let now assume W 4 = 1 2 (p 2 1 + p 2 2 + q 2 1 + q 2 2 ) and E 4 = J. It is easy to see that is the Lie algebra of the Lie group U (2) of 2 × 2 unitary matrices. If we consider u(2) as a subspace of sp(4, R) , then {E 1 , E 2 , E 3 , E 4 } is a basis for u (2). Moreover, the functions W 1 , W 2 , W 3 , and W 4 are a basis for the vector spaces of all quadratic integrals of the harmonic oscillator vector field. The map H : R 4 → R 4 : (q, p) → (w 1 (q, p), w 2 (q, p), w 3 (q, p), w 4 (q, p)), where w 1 = 2W 1 , w 2 = 2W 2 , w 3 = 2W 3 , and w 4 = 2W 4 , is called the Hopf map.

4.
Linear Canonoid Transformations and the harmonic oscillator. When we perform a canonoid transformation of a Hamiltonian system, the integrability or superintegrability of the system are preserved. Indeed, a canonoid transformation is essentially a change of coordinates. Consequently, the existence of intrinsic structures like foliations made of invariant tori (Liouville or complete integrability), or the closure of the finite orbits (maximal superintegrability) is left unchanged. A canonoid transformation may only make these structures more or less evident and easy to handle by allowing the determination, together with the new coordinates, of a new Hamiltonian function and a new symplectic structure for the same dynamical system.
From Prop. 12, it is clear that canonoid and Poissonoid transformations of a Hamiltonian system preserve the functionally independent constants of motion of the system. Therefore, the transformed of a superintegrable system is again superintegrable with the transformed constants of motion. If the transformation is linear, then the degree of the polynomial constants of motion is also preserved by the transformation.
We see below how the two-dimensional harmonic oscillator, which admits the three functionally independent first integrals W 2 , W 3 and W 4 seen above, behaves under linear canonoid transformations.
For our purpose, it is useful that the canonoid transformation of the two dimensional harmonic oscillator leads to a system with a Hamiltonian in one of the two forms . We remark that the manifold where K 1 is defined is the Euclidean plane, while K 2 is defined in the Minkowski plane. Moreover, in the Euclidean case the form of the Hamiltonian is non-restrictive, since any non-degenerate Hamiltonian can be put in this form by a real canonical point-transformation.
A linear canonical point-transformation changes K 2 into P 2 1 −P 2 2 +V . The Hamiltonians W 2 and W 3 are then of the same form of K 2 . Under the same constraints, we apply linear canonoid transformations to the system of Hamiltonian H = 1 2 (p 2 1 + p 2 2 + q 2 1 ), that we can consider as an embedding of the one-dimensional harmonic oscillator in E 2 . This system possesses two evident quadratic first integrals, plus a functionally independent third-one that is not globally defined. We recall that the linear transformation A is canonoid if and only if (7) holds. In order to obtain Hamiltonians of the prescribed form, we must constrain the 2 × 2 submatrix in the lower-right corner of C = −JAJSA −1 to be 1 0 0 1 , or 0 1 1 0 , respectively, where the matrix S is determined by the original Hamiltonian. Moreover, the 2 × 2 submatrices in the upper-right and lower-left corners of C must be equal to zero. We can check if the transformations are canonical thanks to Proposition 9.
With these constraints, we search for canonoid transformations of the isotropic harmonic oscillator H = 1 2 (p 2 1 + p 2 2 + q 2 1 + q 2 2 ). In this case In order to perform computations with a computer-algebra software, we put c 12 = b 21 = 0. With this simplification we find that only one matrix C determines a Hamiltonian of the form K 1 and can be obtained by several different matrices A. The Hamiltonian K 1 is Therefore, the isotropic harmonic oscillator corresponds only to itself by a canonoid transformation of the prescribed type. In particular, we remark that it is impossible to obtain superintegrable anisotropic harmonic oscillators in this way. This is because one of the constants of motion must be of degree higher than two in the momenta or the coordinates [12,13]. If we start from the system of Hamiltonian We find that the canonoid transformations maps (10) into either a Hamiltonian identical to (10), or where k, α 1 and α 2 are constants. In the last case, the point-transformation for some suitable constant k .  By imposing the constraint corresponding to K 2 , we are mapping the Euclidean harmonic oscillator into the Minkowski plane. Let us apply first the canonoid transformation to the isotropic oscillator. We obtain, with the same assumptions on A, that, for the admissible solutions, K 2 is always in the form for some constant k. This is essentially the form of W 2 of the previous section. This type of Hamiltonians corresponds to a well known class of quadratically superintegrable systems of the Minkowski plane, classified as Class II in [8] (in this reference, manifolds and Hamiltonians are considered complex while we limit ourselves to the real case).
We search now for canonoid transformations of the system (10) leading to Hamiltonians of the form K 2 . In this case it is possible to consider the matrix A in full generality. We find that the Hamiltonian of the transformed system, when the transformation is canonoid, is always in the form where k, α 1 and α 2 are constants. After the point-transformation determined by we have K 2 = α 1 α 2 (−P 2 X + P 2 Y ) + kX 2 . We can divide the Hamiltonian by the constant α 1 α 2 = 0 and see that, similarly to the original one, it admits two evident quadratic first integrals and the functionally independent local third integral Y + P Y ln(X + P X ).
In this case the local superintegrable structure remains unchanged. After the computation, we observe that n canonoid transformations such that α 1 α 2 = 0 do not exist. Nevertheless, we can analyze the superintegrability of the system of Hamiltonian (11) in this case. If, say, α 2 = 0, then the system admits two evident quadratic first integrals plus the third-one A computation shows that none of the transformations leading to the last form of K 2 is canonical.

Whittaker's characterization. Since the first edition (1904) of his celebrated
Treatise on Analytical Mechanics [30], E. T. Whittaker characterizes what we call here canonoid transformations. Given a system of ODEs d dt x r = X r (x 1 , . . . , x n , t), r = 1, . . . , n, and a one-form M (x r , t), the absolute and relative integral invariants of the differential equations are defined following Poincaré [24]. We do not need here to recall the definitions of integral invariants (for this, see [30], § §112-116), but only their characterization in modern notation. We have that Indeed, if we consider time-independent systems and if we identify the variational quantities δq i with the differentials dq i , then Σ N i=1 p i δq i becomes the Liouville oneform θ = i p i dq i , and −dθ = ω = dq i ∧ dp i becomes the symplectic form. Hence, by Cartan's magic formula, we have that and thus the vector field X is locally Hamiltonian. Moreover, if the manifold is contractible, thanks to the Poincaré lemma we have for some function H. This means that the system is Hamiltonian, and the previous statement follows in the case of the relative integral invariance condition. For the absolute integral invariance we have and the system is clearly Hamiltonian with Hamilton function i X (θ). Finally, in §136 of [30], the transformations of coordinates (P j (q i , p i ), Q j (q i , p i )), which maintain the Hamiltonian form of (12), our canonoid transformations, are naturally characterized as those for which the form PdQ determines an integral invariant (relative or absolute) of the ODEs. Canonical transformations are defined in the same section of [30].
This characterization provides a simple and direct way to characterize the possible canonoid transformations, or, equivalently, the possible alternative Hamiltonian representations for the field X . Given a system of Hamiltonian H on a symplectic manifold with symplectic form ω, such that i X ω = dH, we can determine another local Hamiltonian structure for the field X whenever we know some non-closed one-form Θ, such that dΘ is non degenerate, satisfying In this case, by (13), we know that, at least locally, i X dΘ = dK for some Hamiltonian function K. A stronger, global, condition is provided if Θ is an absolute integral invariant with dΘ non degenerate. In this case, by (14), Ω = −dΘ is the new symplectic form and the new Hamiltonian K of the system is In both cases, when we can write Θ = P i dQ i for some coordinate system (P i , Q i ), the transformation (p i , q i ) ↔ (P i , Q i ) is canonoid. We remark that, if Θ− p i dq i = df for some function f , then the transformation is the identity.
, the condition L X Θ = 0 becomes a system of first-order PDEs in Θ i (q j , p j ) involving the Hamiltonian H. These last two conditions are not very different from those given for generating functions of canonoid transformations in Cariñena and Rañada [4].
We can call the one-forms Θ such that L X Θ = 0 absolute generators (or global generators) of a canonoid transformation. We call Θ a relative generator (or local generator) of a canonoid transformation when If (p i , q i ) are canonical coordinates, then the Liouville one-form Θ = p i dq i is a relative generator of the identity transformation.
For example, if H is the Hamiltonian of the harmonic oscillator with coordinates (x 1 , x 2 , x 3 , x 4 ) = (q 1 , q 2 , p 1 , p 2 ), that is, Therefore, the absolute generators of canonoid transformations of the harmonic oscillator are characterized by with the evident integrability conditions a non degenerate form with d(J − Θ) = 0, and K = i X Θ = −(p 2 1 + q 2 1 + q 1 p 2 − p 1 q 2 ), that is a first integral of H. Then, the form Θ and the function K provide an alternative Hamiltonian structure for the harmonic oscillator. In the case when H 1 (M, R) is zero, relative generators also determine global Hamiltonian structures.  Let (M, π) be a Poisson manifold, and let X be a locally Hamiltonian vector field on M , that is, for each x ∈ M , there is a neighborhood U of x and locally defined function H U such that X = π · dH U . A diffeomorphism f : M → M is said to be Poissonoid with respect to the vector field X if the transformed vector field f * X is also locally Hamiltonian, that is, for each x ∈ M , there is a neighborhood V of f (x) and a locally defined function K V such that This is equivalent to saying that, for each y ∈ M , there is a neighborhood V of y and a locally defined function K V such that X = (f * π ) · d(f * K V ).

Remark 6.
By Proposition 4 the previous definition means that, for each point x ∈ M , the system of equations associated with X = π · dH U can be written, in splitting coordinates in the neighborhood U of x, as: and the system associated with f * X = π ·dK U can be written, choosing appropriate splitting coordinates in a neighborhood V of f (x), as: so that the transformation f carries the system of Hamilton's equations (16) again into a system of Hamilton's equations (17).
Note that, in the case of a Poisson manifold, not all Poisson vector fields are locally Hamiltonian. This means that, if f is a Poissonoid map then L X (f * π) = 0, but the converse is in general not true. For instance, if π is the trivial Poisson bivector, then any diffeomorphism pushes π to the trivial bivector and L X (f * π) = 0, for any X . On the other hand, the only locally Hamiltonian vector field is the trivial one. If L X (f * π) = 0, we say that the map is weakly Poissonoid with respect to the vector field X . Weakly Poissonoid maps, in general, do not lead to the nice structure described in Remark 6.  (M, π), a diffeomorphism f : M → M is a Poissonoid transformation with respect to X if the transformed field f * X is also Hamiltonian with respect to π, that is, if there is a smooth function K on M such that f * X = π · dK.
Since f is a diffeomorphism, we have that the vector field f * X is Hamiltonian with respect to π if and only if X is Hamiltonian with respect the transformed bivector f * π, i.e. there exists a smooth function K on M such that X = (f * π ) · d(K ).

GIOVANNI RASTELLI AND MANUELE SANTOPRETE
This means that, if f is a Poissonoid transformation for X , then X admits a new and possibly different Hamiltonian structure. If, in addition, π and f * π are compatible, then the vector field X will be biHamiltonian. Under some additional conditions these facts are enough to show that the system has a complete set of integrals in involution.
More explicitly, the existence of the additional Poisson bivector f * π, provides a concrete way of obtaining additional constants of motion from the Casimirs of f * π. Indeed, Proposition 11. Let (M, π) be a Poisson manifold, and let { , } be the Poisson bracket associated with π. Suppose X is a Hamiltonian vector field, so that there exists a function H such that X = π · dH. Suppose that f is a Poissonoid transfomation for X , so that there exists a function K such that X = (f * π ) · d(K ). Then K and any Casimir of the Poisson bivector π are constants of motion of the Hamiltonian system (M, π, H).
Proof. Since f is a Poissonoid transformation for X we have that Suppose C is a Casimir of (f * π) , and that { , } is the Poisson bracket associated to (f * π), then 0 ={C, K } = (f * π)(dC, dK ) = dC, (f * π ) · dK = dC, X = dC, π · dH = π(dC, π · dH ={C, H} Hence, {C, H} = 0, and C is a constant of motion of the Hamiltonian system (M, π, H). The proof that K is a costant of motion is similar.
Another important property is that Poissonoid transformations preserve constants of motion: Proposition 12. Let (M, π, H) be a Hamiltonian system, and let f be a Poissonoid transformation such that f * X H = π · dK, then F is a constant of motion of the transformed system if and only if f * F is a constant of motion for (M, π, H). Proof.
since X K = f * X H . Taking the pull-back of the above yields The proof follows.

Remark 8.
Since the transformation f is a diffeomorphism, functionally independent constants of motion are sent into functionally independent constants of motion. In the case when the Poisson structure coincides with a symplectic structure, Proposition 12 applies to canonoid transformations. where I 1 , I 2 , I 3 are the principal moments of inertia of the rigid body. The corresponding Hamiltonian vector field, given by X = π · dH is: Consider the diffeomorphism f : M → M such that (n 1 , n 2 , n 3 ) = f (m 1 , m 2 , m 3 ) defined by the equations n 1 = am 1 , n 2 = bm 2 , n 3 = cm 3 , where a, b and c are non-zero constants. The pushforward of X can be expressed as and one possible Hamiltonian, obtained from f * X = π · dK, is Pulling back K we get f * K = − 1 2 (m 2 1 + m 2 2 + m 2 2 ). Pulling back the bivector π yields
Example 6.3 (Euler's equations on so * (4)). Here we use the same notations as in [9]. The manifold M = so * (4) is six dimensional. Since so(4) is isomorphic to the space of 4 × 4 skew-symmetric matrices, identifying so * (4) with so(4) we can write any element of so * (4) as where E ij denotes the elementary matrix whose (i, j) entry is 1.
The Euler-Manakov equations for the rigid body on (so * (4), π) are the Hamiltonian equations with the following quadratic Hamiltonian where the coefficients a ij can be written as defines an isomorphism of the Lie algebras (R 3 , ×) and (so (3), [, ]). If we identify the Lie algebra e(3) with its dual, in these notations, the Lie-Poisson tensor has the form This Poisson tensor has the following quadratic Casimirs: Hamilton's equations corresponding to a quadratic Hamiltonian are called Kirchhoff equations.
A famous integrable case of the Kirchhoff equations was discovered by Clebsch and it is characterized by the Hamiltonian

The Hamiltonian vector field in this case is
and an additional integral of motion is Note that a linear change of variables transforms the Euler equations on so * (4) to equations on e * (3), which in the case of a positive definite quadratic Hamiltonian are the Kirchhoff's equations describing the motion of a rigid body in an ideal fluid [2]. Thus, in principle, the Poisson bivector above and the Poissonoid transformation we find below could be obtained from the previous example. However, we prefer to find them with a direct computation. In this case, it is easy to verify that X = π · dH 1 = ( η) · d(− 1 2 C 1 ). It can also be shown that this Poisson structure is compatible with the original one. Hence, the Clebsch system is biHamiltonian.
In analogy with the previous examples, we search now for a transformation φ = f −1 such that (we consider here − 1 2η instead ofη for computational convenience) and, consequently, X = (f * π ) · d(C 1 ).
While in all the previous examples f was a simple rescaling of the old coordinates, here we must assume that f , and thus φ, are linear and homogeneous in (p 1 , p 2 , p 3 , m 1 , m 2 , m 3 ) and (F 1 , . . . , F 6 ) = f (p 1 , p 2 , p 3 , m 1 , m 2 , m 3 ) respectively, so that with a k j constants. The previous relations allow to write the components ofη with respect to the (φ j ) as functions of the (F j ), therefore, we can solve the equation (18) without inverting the transformation φ. Substituting φ = f −1 and X = F in (4), and using (19), yields: Therefore, condition (18) written explicitly in local coordinates is jk (φ(F )) = a j r a k s π rs (φ −1 (φ(F ))) = a j r a k s π rs (F ), that can also be written as This last equation can be solved for the a j k giving for example the following linear transformation where a, are real parameters, If ω 1 > ω 2 > ω 3 ≥ 0 then a positive value of can always be found such that the transformation is real. In the new coordinates we have 7. Infinitesimal Poissonoid transformations and symmetries. Let (M, π, X ) be a locally Hamiltonian system with a locally defined Hamiltonian function H, and f t a one-parameter group of Poissonoid diffeomorphisms, then (f t ) * X = π · dK t . This expression can be rewritten as . Let ξ be the infinitesimal generator of f t . Differentiating the previous equation with respect to t yields where we used the definition of Lie derivative andK is the derivative of K with respect to t computed at t = 0. Hence, where F = −K. A vector field ξ satisfying the equations above is called a infinitesimal Poissonoid transformation of (M, π, X ). In [3] canonoid transformations are studied using cohomology techniques. If (M, π, X ) is a locally Hamiltonian system, then, in analogy with [3] one can introduce twisted boundary and coboundary operators defined as follows: where d is the usual de Rham differential and i X denotes contraction.
Proposition 13. Suppose (M, π, X ) is a locally Hamiltonian system with (locally defined) Hamiltonian H, then X F is a Hamiltonian infinitesimal symmetry of (M, π, X ) if and only if d X F ∈ ker(π ).
Proof. Let F ∈ C ∞ (M ), and let X F = π · dF be its Hamiltonian vector field. Let α be an arbitrary one form, then the preceding equation can be written as Taking the Lie derivative of the left hand side yields Taking the derivative of the right hand side yields L X ( α, π · dF ) = L X (π(α, dF )) = (L X π)(α, dF ) + π(L X (α), dF ) + π(α, L X (dF )) = π(L X (α), dF ) + π(α, L X (dF )) = L X (α), π · dF + α, π · (L X (dF )) , because X is locally Hamiltonian. Since α is arbitrary, comparing (21) and (22) gives Moreover, by Cartan's magic formula Thus [X , X F ] = 0 if and only if d X F is in ker(π ). Since X is locally Hamiltonian we also have that L X H = 0 If π is non-degenerate and H 0 X (M ) = {F ∈ C ∞ (M )|d X F = 0} is the zero cohomology group of d X , then H 0 X (M ) coincides with the set of Hamiltonian infinitesimal symmetries of (M, π, X ), reproducing the result given in [3] for the symplectic case. This result suggests that the cohomology approach introduced in [3], is not well adapted to the Poisson case. It may be possible to give a nice cohomological interpretation of symmetries in this case by using certain cohomology groups associated with a foliated space, called tangential cohomology groups (see [21]) and references therein).
Let α be a differential one-form on (M, π). We denote by α the vector field π (α).
Lemma 7.2. Let (M, π, X ) be a locally Hamiltonian system. The vector field β = d X F is β = [X , X F ], where X F is the Hamiltonian vector field of F .
Proof. Equation (23) yields β = [X , X F ] = π · (L X (dF )) = π · d X F = π · β. Proposition 14. Let (M, π, X ) be a locally-Hamiltonian system and let β be a one-form on M . Then the vector field β = π · β is an infinitesimal Poissonoid transformation if and only if d X β = dα, where α ∈ ker π . In particular, if d X β = 0, then β is an infinitesimal Poissonoid transformation. If π is non-degenerate then the vector field β is an infinitesimal Poissonoid transformation if and only if d X β = 0.
This result generalizes the well known fact that infinitesimal canonical transformations are generated by closed forms in the de Rahm cohomology (that is the basis of the theory of generating functions) and the fact that infinitesimal canonoid transformations are generated by closed forms in the twisted cohomology introduced in [3] to the case of Poissonoid transformations. In this case, however, not all the infinitesimal Poissonoid transformations are generated by d X -closed differential forms.
It seems possible to state the proposition above more elegantly by introducing differential forms that are tangential to the foliation, see [21] and references therein for possible definitions of these forms. 7.1. Master Symmetries. Let (M, π, X ) be a locally-Hamiltonian system. Recall that an infinitesimal symmetry of X is a vector field ξ such that [ξ, X ] = 0. A master symmetry for X is a vector field ξ such that The last condition in the equation above can also be written as L m+1 X (ξ) = 0. Neither master symmetries nor Poissonoid infinitesimal transformation do, in general, generate constants of motion. However, the next proposition shows that some special infinitesimal Poissonoid transformations generate constants of motion.
Proposition 15. Let ξ be an infinitesimal Poissonoid transformation of (M, π, H), such that the relationship [ξ, X H ] = π ·dF = 0 is satisfied globally and L [ξ,X H ] H = 0. Then F is a constant of motion of (M, π, H) and ξ is a master symmetry of degree 1.
Note that, in particular, If T is the generator of constants of motion of degree m, then L m+1 X T = 0, and thus d(L m+1 X T ) = 0. It follows that L m+1 X (ξ) = 0, and thus, if L X (ξ) = 0, . . . , L m X (ξ) = 0, ξ is a master symmetry of degree m. The converse is clearly not true if π is degenerate.
The following proposition shows that Poissonoid transformations are very general: every master symmetry, and also every infinitesimal symmetry of the vector field can be generated using Poissonoid transformations.
Proposition 16. Let (M, π, X ) be a locally-Hamiltonian system. Suppose that ξ is a master symmetry of degree m. Then the vector field L m X (ξ) is an infinitesimal Poissonoid transformation.
Proof. Since ξ is a master symmetry of degree m, we have Proposition 17. Let (M, π, X ) be a locally-Hamiltonian system and let β be a one-form on M . Suppose the vector field β = π · β is an infinitesimal Poissonoid transformation such that d X β = 0. Then β is master symmetry of degree m ( m ≥ 1) if and only if i X β is the generator of a Hamiltonian master symmetry of degree m − 1.
Proof. β is a master symmetry of degree m if and only if L m+1 X β = 0, and L X β = 0, . . . , L m X β = 0.