KIRILLOV STRUCTURES AND REDUCTION OF HAMILTONIAN SYSTEMS BY SCALING AND STANDARD SYMMETRIES

. In this paper, we discuss the reduction of symplectic Hamiltonian systems by scaling and standard symmetries which commute. We prove that such a reduction process produces a so-called Kirillov Hamiltonian system. Moreover, we show that if we reduce ﬁrst by the scaling symmetries and then by the standard ones or in the opposite order, we obtain equivalent Kirillov Hamiltonian systems. In the particular case when the conﬁguration space of the symplectic Hamiltonian system is a Lie group G , which coincides with the symmetry group, the reduced structure is an interesting Kirillov version of the Lie-Poisson structure on the dual space of the Lie algebra of G . We also discuss a reconstruction process for symplectic Hamiltonian systems which admit a scaling symmetry. All the previous results are illustrated in detail with some interesting examples.

1. Introduction 1.1.Physical motivation.The analysis of symmetries is one of the most important tools in theoretical physics.Usually, the formulation of a physical theory is given in terms of a variational principle and its associated symplectic Hamiltonian description.In this context, one typically looks for "standard symmetries", that is, symmetries which preserve the symplectic form and the Hamiltonian function.Among other things, this approach leads to Noether's theorem and its generalization and the Marsden-Weinstein theory of reduction of the system by the action of a symmetry group (see the classical books and monographs by Marsden and collaborators [1,37], Libermann and Marle [34] or Olver [43]).
Recently, there has been a growing interest in the physical literature in considering "non-standard symmetries", that is, symmetries of the physical system that do not necessarily preserve the symplectic structure.This is motivated mainly by the so-called scaling symmetries and by a well-known philosophical argument according to which any minimal description of the universe should avoid introducing a global scale into the picture, that is, it should be scale-invariant [30,44].In this context, the theory of "shape dynamics" aims to rephrase our best description of the universe (general relativity) in a completely scale-invariant fashion [5,39].This has led already to remarkable results that defy the way we understand the (classical) dynamics of the universe.For instance, the scale-reduced cosmological and black hole systems can be continued in some cases through the corresponding singularities [33,40,49].Moreover, it has been further argued that the apparent dissipative nature of the scalereduced systems may have important consequences for topics such as the origin of the arrow of time and the formulation of quantum mechanics through unitary operators [6,30,50].
Interestingly, the reduction of a symplectic Hamiltonian system by a scaling symmetry produces a contact Hamiltonian system, which have been the subject of intensive study recently for their use in the description of e.g.dissipative, thermostatted and thermodynamic systems (see e.g.[9, 10, 13, 16, 17, 21-24, 47, 51] and the references therein).This intuition was first put forward in [48] and then formalized more precisely in the recent work [12], where a thorough mathematical investigation of the role of scaling symmetries in symplectic Hamiltonian systems has been performed.Moreover, the relationship with the geometry of the blow-ups used in celestial mechanics has also been highlighted, together with the connection with other geometric structures [8,41].
However, so far the study of the joint reduction by scaling and standard symmetries has not been considered in depth, at least from the mathematical perspective.Moreover, the case in which the reduced manifold is nonorientable, which seems to be the important case for the resolution of singularities in general relativity [33,40,49], has been elusive of a fully-fledged mathematical description (although, see [14,25,36]).Finally, from the point of view of comparing the resulting physical theories, it is also crucial to highlight how to reconstruct the "original" symplectic system from the reduced one.
In this work we perform a detailed mathematical analysis of all the above points.To give a feeling of the objects involved in our constructions, in the remainder of this introduction we provide a high-level description of the most important tools and results.
1.2.Standard Lie symmetries for Kirillov Hamiltonian systems.A Kirillov structure on a real line bundle is a Lie algebra structure [•, •] on the space of sections of the dual bundle such that, if we fix a section h on this line bundle, the operator [•, h] is a derivation.Thus, every section of the dual line bundle defines a vector field on the base manifold which is called the Hamiltonian vector field associated with the section.So, a Kirillov Hamiltonian system is a Kirillov structure on a real line bundle plus a section of the dual bundle (the Hamiltonian section).
Examples of Kirillov structures may be produced from symplectic, Poisson and Jacobi structures, contact 1forms and contact structures (that is, distributions of corank 1 which are maximally non-integrable).Apart from the last case, in the other previous examples the real line bundle is trivial and the sections of the dual bundle are just C ∞ functions on the base manifold.Anyway, as we show in this paper, there exist interesting examples of Kirillov structures for which the real line bundle is not trivial.In particular, those in which the base space of the line bundle is the projective bundle associated with a vector bundle (for more details on Kirillov structures, see for instance, [27,28,31,32,36]).
On the other hand, it is well-known that dynamical systems (in particular, mechanical systems), which are invariant under the action of a symmetry Lie group, have received a lot of attention from researchers in mathematics and physics.For this reason, in this paper we introduce the notion of a standard Lie symmetry for a Kirillov Hamiltonian system.It is a principal representation of a Lie group on the line bundle such that the dual representation preserves the Kirillov structure and the Hamiltonian section is equivariant.A Lie group of symplectic (resp.Poisson, contact or Jacobi) Hamiltonian symmetries is a particular example of a standard Lie symmetry for the corresponding Kirillov Hamiltonian system.Moreover, for a standard Lie symmetry on a Kirillov Hamiltonian system, the space of orbits of the action on the line bundle is again a line bundle.In fact, in the particular case when the Kirillov structure is Poisson (or Jacobi), we have a reduced Poisson (or Jacobi) structure.This is well-known in the theory of Poisson (or Jacobi) reduction (see, for instance, [38,42]).
1.3.Scaling symmetries for Poisson Hamiltonian systems.In [12] the authors introduce the notion of a scaling symmetry for a symplectic Hamiltonian system and they exhibit several examples where such a symmetry is present (see also [4,11,48]).
The previous notion may be extended for the more general class of Poisson Hamiltonian systems as follows.It is a principal action Φ : R × × P → P of the Lie group R × (with R × = R + or R × = R − {0}) on the Poisson manifold (P, Π) such that ∧ 2 T Φ • Π = sΠ • Φ, H • Φ = sH, for all s ∈ R × , where H : P → R is the Hamiltonian function.In the particular case when P is a symplectic manifold S, it is proved in [14,25] that the space of orbits C = S/R × admits a contact structure.In addition, the homogeneous function H on S induces a section of the dual bundle over C to the Kirillov line bundle in such a way that we have a reduced contact Hamiltonian system (see [25,36]).
1.4.Our motivation.As we mentioned before, many symplectic Hamiltonian systems admit scaling symmetries.However, they do not only admit such symmetries, typically they also have standard Lie symmetries.In addition, the scaling and the standard Lie symmetries usually commute.So, one may reduce the dynamics by both types of symmetries, and some natural questions arise: • What is the nature of the reduced system?• If we reduce first by the scaling symmetries and then by the standard ones, is it the same as doing it the other way around?• Is it possible to obtain the dynamics of the original symplectic Hamiltonian system from the dynamics of the reduced system via a suitable reconstruction process?
In this paper, we will provide answers to these questions.
1.5.The results of the paper.For a symplectic Hamiltonian system with compatible scaling and standard Lie symmetries (that is, they commute), we will develop two reduction processes: • In the first reduction process, we start with the standard symmetry and then we apply the scaling symmetry.
In this case, the first reduced system is a Poisson Hamiltonian system endowed with a scaling symmetry.
The reduction of such a system by this scaling symmetry produces a Kirillov Hamiltonian system (see Theorem 4.3).
• In the second reduction process, we use the scaling symmetry and then the standard symmetry.In this case, the first reduced system is a contact Hamiltonian system endowed with a standard Lie symmetry.The reduction of the latter by this standard symmetry produces again a final Kirillov Hamiltonian system (see Theorem 5.4).In fact, the reduction of a general Kirillov Hamiltonian system by a standard Lie symmetry is again a Kirillov Hamiltonian system (see Theorem 5.2).• We also prove that the final reduced Kirillov Hamiltonian systems obtained in both processes are Kirillov equivalent (see Theorem 6.1).
A contact 1-form on a (2n + 1)-dimensional manifold C is a 1-form η such that η ∧ (dη) n defines a volume 1-form on C. We remark that a manifold with a contact 1-form is orientable and has a distinguished vector field R ∈ X(C), the Reeb vector field, characterized by the conditions i R dη = 0 and i R η = 1.
The Reeb dynamics can be seen as the one induced by a Hamiltonian vector field on C. In fact, if H : C → R is a smooth function on C, the Hamiltonian vector field X η H ∈ X(C) of H is characterized by these two conditions (1) i The Reeb vector field is just the Hamiltonian vector field for the constant function H = 1.
In the following example we show a manifold endowed with a contact 1-form obtained by a reduction process.
Example 2.1 (The spherical cotangent bundle of a Riemannian manifold).Let (Q, g) be an n-dimensional Riemannian manifold and 0 Q the zero section of the cotangent bundle τ * Q : consider the action of the multiplicative group of the positive real numbers R + given by (2) φ : or equivalently, where g denotes here the corresponding metric on T * Q.
In the particular case when Q is R n+1 , with the flat Riemannian metric, we have that the spherical cotangent bundle is with S n the n-sphere in R n+1 .[7,45,46]).
We remark that the regular and singular Marsden-Weinstein reduction of the spherical cotangent bundle have been discussed some years ago [19,20].In fact, this reduction process is a particular case of the more general Marsden-Weinstein contact reduction which has been intensively discussed by several authors [2,18,26,29,54].
A contact 1-form is a particular case of a Jacobi structure.A Jacobi manifold M ( [32,35]) is endowed with a pair (Π, E) ∈ V 2 (M ) × X(M ), where Π is a 2-vector field and E is a vector field on M such that •]] being the Schouten-Nijenhuis bracket on M .Associated with a Jacobi manifold (M, (Π, E)) we have a Jacobi bracket, given by ( 5) which is a Lie bracket on the space of functions on M such that In fact, a Jacobi bracket on the space of functions C ∞ (M ) defines a Jacobi structure (Π, E) satisfying (5).
Note that we have a vector field on M , the Hamiltonian vector field associated with f 2 , such that In terms of the Jacobi estructure, this vector field is given by ( 7) If E = 0 we recover the notion of a Poisson bracket on the space of functions on M and (M, Π) is a Poisson manifold.
For a manifold C with a contact 1-form η, the Jacobi structure is , where R is the Reeb vector field associated with η and ♭ η : Moreover, the Hamiltonian vector field defined in ( 1) is just the corresponding Hamiltonian vector field X {•,•}M f associated with the Jacobi structure (Π η , E η ) (see [35]).
Example 2.2 (continuing Example 2.1).In the case of the spherical cotangent bundle of a Riemannian manifold (Q, g), we consider the differentiable function κ g : , where Π ωQ is the Poisson structure induced by the symplectic structure ω Q on T * Q.
On the other hand, contact 1-forms are also a particular kind of more general structures which are not, in general, Jacobi structures.
A contact structure on a (2n + 1)-dimensional smooth manifold C is a distribution D on C of codimension 1 which is maximally non-integrable, i.e. for all x ∈ C, there is an open neighborhood U of x such that the distribution D on U is given by the annihilator < η U > o of the vector subbundle of T * C generated by a contact 1-form η U on U , that is In this case, the pair (C, D) is a contact manifold.
It is clear that if C has a global contact 1-form, the pair (C, D =< η > o ) defines a contact manifold.But in general, a contact structure on C may be not defined by a global contact 1-form on C as the following example proves.
Example 2.3 (The projective cotangent bundle of a manifold).Let Q be an n-dimensional manifold and 0 Q the zero section of the cotangent bundle τ * Q : we consider the action of the multiplicative group R − {0} given by (8) φ Its infinitesimal generator ∆ Q is the Liouville vector field on Remark 2.4.The notion of projective bundle P(V ) may be defined for an arbitrary vector bundle τ : V → Q as the quotient bundle induced by the action on where 0 Q is the zero section of τ : V → Q.
A particular case is when Q is a point and V is the dual of a Lie algebra g.In this case, the base space of the projective bundle p : g * − {0} → Pg * is just the projective space Pg * .If λ Q is the Liouville 1-form on T * Q and p : (T * Q − 0 Q ) → P(T * Q) is the quotient projection, using (3), one can prove that the distribution of co-rank 1 If D denotes its projection, then (P(T * Q), D) is a contact manifold.
A simple example of this kind of contact manifolds is when Q is a Lie group G.In this case, the cotangent bundle T * G may be left trivialized to the trivial vector bundle G × g * → G, where g is the Lie algebra of G.Under this identification, the action φ is just (s, (g, µ)) → (g, sµ).
Then, the quotient bundle is p = Id G × p : G × (g * − {0}) → G × Pg * and the contact structure is the distribution on G × Pg * given by for all g ∈ G and µ ∈ g * − {0}.Here L : G × G → G denotes the left action of the Lie group G on itself.
In the particular case when G = R n+1 , the projective cotangent bundle P(T * R n+1 ) can be identified with the cartesian product R n+1 × P n (R), where P n (R) is the real projective space of dimension n.This space is nonorientable when n is even and therefore, P(T * R n+1 ) does not admit a global contact 1-form.
Contact and Jacobi structures are special examples of more general structures: Kirillov structures (see [32], and also [14,25,27]).Definition 2.5.A Kirillov structure on a manifold K is a real line bundle π L : a vector field on K.The vector field X The line bundle (L * , π L * , K) with the bracket [•, •] L * on the space of sections of π L * is, in Marle's terminology [36], a Jacobi bundle.This kind of structures are essentially equivalent to the conformal Jacobi structures studied in [15].
When the line bundle π L : L → K is trivial, i.e.L ∼ = K × R, the sections of π L * can be identified with smooth functions on K.Under this identification, the local Lie algebra This means that {•, •} K is a Jacobi bracket, whose associated Jacobi structure (Π, E) is given by Conversely, every Jacobi manifold (K, {•, •} K ) defines a Kirillov structure on the trivial line bundle π : K × R → K. Therefore, Jacobi structures are just trivial Kirillov structures.
In the case of a contact manifold (C, D), consider the line bundle with total space the annihilator bundle D o of D, π D o : D o → C of D, which is, in general, not trivial.Using this line bundle and the representation which defines the symplectic structure ω S = −dλ S .This symplectic structure is homogeneous with respect to the R × -action φ S : R × × S → S on S, i.e.There is a one-to-one correspondence between the sections of π In the previous examples, the reduction processes are the fundamental tool to obtain contact structures from symplectic structures.Now, we will show this process for a general symplectic Hamiltonian system, which was discussed in [25], and then we will present some examples.We begin by recalling the notion of scaling symmetries [12] for this kind of dynamical systems.Definition 3.1.Let (S, ω) be a symplectic manifold and H : S → R a function on S. A scaling symmetry for the dynamical system (S, ω, H) is a principal action φ : R × × S → S of the multiplicative group R × (with Note that if ∆ ∈ X(S) is the infinitesimal generator of the scaling symmetry, then In fact, if R × is connected (that is, R × = R + ), then the previous conditions are equivalent to the fact that the principal action φ is a scaling symmetry.
An immediate consequence of the existence of a scaling symmetry is that the symplectic structure is exact, that is, ω = −dλ with λ = −i ∆ ω.Moreover, the 1-form λ is homogeneous, i.e. (φ s ) * λ = sλ, and if Π ω is the Poisson bi-vector induced by ω, then Π ω satisfies the following relation where ∧ 2 T φ s : ∧ 2 T S → ∧ 2 T S is the vector bundle isomorphism induced by the diffeomorphism φ s : S → S.
Now, we will develop the reduction process with the scaling symmetry φ.
Denote by C := S/R × the corresponding quotient manifold and by p S : S → C its quotient projection.Then, we may consider the distribution which is p-projectable and the corresponding distribution D on C, which is a contact structure.
Denote by [•, •] (D o ) * the Kirillov bracket on the space of sections of the line bundle π (D o ) * : (D o ) * → C characterized by (10).On the other hand, from the homogeneity of H : S → R with respect to the scaling symmetry, we have a section h of h H given as in ( 9) is just the p-projection on C of the Hamilton vector field X ω H .The following diagram summarizes this reduction process (see [25], for more details on this reduction process).
pS H 8 8 q q q q q q q q q q q q (C, D, X Now, we will exhibit two examples of contact dynamical systems induced by a scaling reduction process. Example 3.2 (The 2d harmonic oscillator and the spherical cotangent bundle).Consider the manifold Then, under this identification the space ), we have that the local expression of the standard symplectic form ω Q and the corresponding Poisson bi-vector Π ωQ on R + × S 1 × R + × S 1 are respectively Now, we consider the symplectic Hamiltonian system (T * Q, ω Q , H) of the harmonic oscillator where, under the identification ( 12), H : T * Q → R is the Hamiltonian function given by ( 14) with r, r ′ ∈ R + .In this case the dynamics is given by the Hamiltonian vector field We consider the action of Note that it defines a scaling symmetry, since On the other hand, the diffeomorphism transforms the generator ∆ of the R + -action on R + × S 1 × R + × S 1 into the vector field 1 2 ρ∂ ρ .The inverse of this map is (ρ, θ, ρ ′ , θ ′ ) → (ρ, θ, ρρ ′ , θ ′ ).Then, we have that: • The contact 1-form under this identification is given by The Reeb vector field associated with this contact 1-form is From the homogeneity of the Poisson structure {•, •} ωQ with respect to the symplectic form From this fact and using the local expression of Π ωQ with respect to the coordinates (ρ, θ, ρ ′ , θ ′ ), we obtain the Jacobi bracket associated with the contact structure defined by η Therefore, the Jacobi structure is given by ( 16) • The reduced Hamiltonian function H is the function ).
• The reduced vector field on , which is just the contact Hamiltonian vector field of the restriction H |R + ×S 1 ×S 1 with respect to the contact 1-form η or, equivalently, the Jacobi Hamiltonian vector field of In the previous example the Hamiltonian function H induces a function H |C on the reduced space C.However, in general, we do not necessarily have a function on the reduced space, as the following example proves.
Example 3.3.The projective cotangent Hamiltonian system deduced from a standard linear Hamiltonian system.Let Y ∈ X(Q) be a vector field on the manifold Q of dimension n.We denote by Y ℓ : where Y (q) = Y i (q)∂ q i .We remark that the linearity of Y ℓ implies its homogeneity, i.e.
with respect to the action given in (8).
• The contact distribution D on p(U i0 ) is just The particular case of a Lie group.When Q is a Lie group G and the vector field Y on G is left-invariant, we have (see Example 2.3): • The vector field Y is given by Y (g) = T e L g (ξ), with ξ an element of the Lie algebra g of G.
• The linear function • The contact structure is the distribution on G × Pg * given by Here p : g * −{0} → Pg * is the corresponding quotient map determined by the scaling symmetry on g * −{0}.• The fiber of the line bundle [1]).Here [•, •] g is the Lie algebra structure on g.Then, the Hamiltonian vector field where ξ ℓ is the restriction to g * − {0} of the linear function ξ ℓ : g * → R induced by ξ and {•, •} g * −{0} is the restriction to functions on g * − {0} of the Lie-Poisson bracket on g * .We recall that this bracket is characterized by ), with α ∈ g * and ξ i ∈ g (for more details, see [1]).
The reduced vector field after this reduction is just (Y, A more explicit (local) expression of the vector field X h ξ ∈ X(Pg * ) may be obtained as follows.For each ν ∈ g − {0} one can consider the coordinate open neighborhood p(U ) of Pg * with U = {α ∈ g * /ν ℓ (α) = α(ν) = 0}.On p(U ) the typical local coordinates in Pg * have the form r(ζ, ν) characterized by Moreover, using (20) and ( 21), we deduce that Given the above facts, it is natural to ask if it is possible to extend the previous reduction to a Poisson Hamiltonian system, not necessarily symplectic.The following result gives an affirmative answer to this question.Before that, we introduce the notion of scaling symmetry for this kind of systems.Definition 3.4.If (P, Π, H) is a Poisson Hamiltonian system on the Poisson manifold (P, Π), a scaling symmetry for (P, Π, H) is a principal action φ P : R × ×P → P of the multiplicative group R × (with R × = R + or R × = R−{0}) on P such that the Poisson structure Π and the function H are homogeneous with respect to the action φ P , that is, , where ∧ 2 T φ P s : ∧ 2 T P → ∧ 2 T P is the vector bundle isomorphism induced by φ P s : P → P .
The conditions in (23) are equivalent to the following ones where {•, •} P is the Poisson bracket of functions on P.
We remark that (23) implies that the Poisson structure Π and the Hamiltonian function H satisfy (see [15,36]) where ∆ P is the infinitesimal generator of φ S .Moreover, if R × is connected (that is, R × = R + ) the previous conditions are equivalent to (23).In addition, in the case of a symplectic manifold (S, ω), the condition with Π ω the Poisson structure induced by ω, is equivalent to (φ P s ) * ω = sω.We have the following important result.Theorem 3.5.Let p P : P → K = P/R × be a principal R × -bundle with total space a homogeneous Poisson manifold (P, Π).If π L : L → K is the line bundle associated with the principal bundle p P (see Appendix A), then: a) There is a one-to-one correspondence between homogeneous functions H : P → R and sections h H : where Proof.For a proof of a) see Appendix A.
If H 1 , H 2 are two homogeneous functions then, , and, using (24), we deduce that is homogeneous.Thus, the Poisson bracket {•, •} P is closed for homogeneous functions with respect to ∆ P .

Using this fact and Proposition
This bracket was described (up to the sign) in [14] (Theorem 3.2).Using the fact that the Poisson bracket {•, •} P defines a Lie algebra on the space of functions on P and (25), we deduce that [•, •] L * is a Lie bracket.Moreover, for a C ∞ function f : K → R, from the properties of the Poisson bracket {•, •} P , we have that ( 26) On the other hand, using the homogeneity of Π and H h2 , we deduce that From ( 26) and ( 27), we have that and consequently (see ( 9)) we have a Kirillov structure on the space of sections of π L * : L * → K and the symbol of [•, h] L * is just the p P -projection on K of the Hamiltonian vector field . This proves b) and c).
Finally, from (27) and using that X Remark 3.6.In [36] Marle proves that if π L : L → K is a line bundle endowed with a Kirillov structure -(L * , π L * , K) is a Jacobi bundle in his terminology -and h : K → L * is a section of π L * , then one can induce a Poisson structure Π on L * (which is homogeneous with respect to the negative of the Euler vector field ∆ on L * ), a differentiable function H : P := (L * − 0 L * ) → R and a vector field X on L * such that: • The restriction of X to P is just the Hamiltonian vector field induced by Π and H.
• The vector field X projects on a vector field X h on K (see Theorem 4.3 and Proposition 4.7 in [36]).Therefore, if the flow of ∆ induces a principal action on P, then we have a Poisson Hamiltonian system (P, Π, H) with a scaling symmetry in such a way that the corresponding reduced Kirillov Hamiltonian system is just the original system.So, Marle's result may be considered as a converse of Theorem 3.5.
Remark 3.7.In [53] (see Theorem 2.2.6 of [53]), the authors obtain a one-to-one correspondence between Atiyah (l, m)-tensors on a line bundle and homogeneous (l, m)-tensors on its slit dual bundle (the dual bundle with the zero section removed).Using this general result, one could prove that there exists a one-to-one correspondence between Kirillov structures on the line bundle and homogeneous Poisson structures on its slit dual bundle (see Example 2.4.2 in [53]).Anyway, in order to have our paper more self-contained, we have included a direct and simple proof of the items a), b), c) and d) of Theorem 3.5.
The following diagram summarizes Theorem 3.5 Reduction of symplectic Hamiltonian systems using first standard symmetries and then scaling symmetries In this section, we will discuss the reduction of symplectic Hamiltonian systems which are invariant under the action of a symmetry Lie group and, in addition, admit a scaling symmetry.The standard and the scaling symmetries will be compatible in the following sense.Definition 4.1.Let (S, ω, H) be a symplectic Hamiltonian system.Suppose that φ S : R × × S → S is a scaling symmetry on (S, ω, H).Additionally, suppose that we have a Lie group G and a G-principal bundle ℘ S : S → S/G such that the corresponding action Φ S : G × S → S on the symplectic manifold S satisfies: , for all x ∈ S and g ∈ G.
(iii) The symplectic and the scaling actions commute, that is, Φ S g • φ S s = φ S s • Φ S g , for all s ∈ R × and g ∈ G.
In this case we say that the dynamical system (S, ω, H) admits a scaling symmetry φ S : R × × S → S and a symplectic G-symmetry Φ S : G × S → S which are compatible.
Note that the previous conditions (i) and (ii) imply that ( 28) where ξ S is the infinitesimal generator of the action Φ S associated with an element ξ of the Lie algebra g of G and Π ω is the Poisson bi-vector on S induced by the symplectic structure ω.If G is connected, then the conditions (i) and (ii) are equivalent to (28).
4.1.The first step: Reduction by standard symmetries.It is well-known (see [38]) that the symplectic structure on S induces a Poisson bracket {•, •} P on the quotient manifold P := S/G characterized by ( 29) with f i ∈ C ∞ (P ), where {•, •} S is the Poisson bracket induced by the symplectic structure ω on S. Consequently, the Poisson structure Π P on P and the Poisson structure Π ω induced by the symplectic structure ω are related as follows In addition, from the G-invariance of H, there is a reduced Hamiltonian function Moreover, the Hamiltonian vector field X ω H ∈ X(S) is ℘ S -projectable and its projection is just the Hamiltonian vector field The following diagram summarizes this first reduction process On the other hand, using that Φ S g • φ S s = φ S s • Φ S g , for all s ∈ R × and g ∈ G, the R × -action φ S induces an action φ P : R × × P → P characterized by (32) φ P s (℘ S (x)) = ℘ S (φ S s (x)), for all x ∈ S and s ∈ R × .
Then, we have Proposition 4.2.φ P is a scaling symmetry for the Poisson Hamiltonian system (P, Π P , H G ).
Proof.Given s ∈ R × , using (30) and (32), it follows that Now, since φ S is a scaling symmetry for the symplectic manifold (S, ω), we deduce that and, using again (30), we obtain that This implies that On the other hand, from (31) and (32), it follows that and, since H is a homogeneous function for the action φ S , we deduce that where for the last equality we use again (31).This implies that which ends the proof of the result.Now, we may apply the scaling reduction process.

4.2.
The second step: Reduction by scaling symmetry.Consider the Poisson Hamiltonian system (P, Π P , H G ) obtained in the previous subsection by reduction from the symplectic Hamiltonian system (S, ω, H).In the second step of the reduction process we will apply Theorem 3.5 to the Poisson Hamiltonian system (P, Π P , H G ) and the scaling symmetry φ P : R × × P → P.
The complete reduction process is described in the following theorem.
Theorem 4.3.Let (S, ω, H) be a symplectic Hamiltonian system with compatible scaling symmetry φ S : R × ×S → S and symplectic G-symmetry Φ S : G × S → S, G being a Lie group.Then: (1) The multiplicative group R × acts on the Poisson manifold P = S/G such that the corresponding quotient map p P : P → P/R × is a R × -principal bundle.Moreover, if π L : L → K = P/R × is the line bundle associated with p P : P → K = P/R × , then the homogeneous function P ) homogeneous functions on P.
(3) The Hamiltonian vector field X ω H is (p P • ℘ S )-projectable on K and its projection is the symbol The following diagram illustrates both reduction processes together.
where (r, θ, r ′ , θ ′ ) are polar coordinates on One can also easily check that [ξ S , ∆ S ] = 0 and thus, since the multiplicative group R + and S 1 are connected, the two symmetries commute.Therefore, the corresponding actions are compatible and we can apply Theorem 4.3.In order to highlight all the mechanisms involved, we will proceed by steps and indicate the main derivations.
In the first step, with the S 1 -symmetry, the reduced objects are: • The reduced space: We perform the reduction by the standard symmetry, obtaining the Poisson system (P, Π P , H G ). Firstly, we have that the symplectomorphism transforms ξ S into ∂ α .Using this identification, the quotient manifold (R + × S 1 × R + × S 1 )/S 1 is just and the reduced Poisson structure on P is given by (see ( 13)) where (r, r ′ , α) are local coordinates on R + × R + × S 1 .• The reduced Hamiltonian function: The reduced Hamiltonian function is • The reduced dynamics: The corresponding Hamiltonian vector field on P is just • The scaling symmetry on the reduced space: The projection on P of the scaling symmetry ∆ S is which generates the scaling action φ P : R + × (R + × R + × S 1 ) → (R + × R + × S 1 ) given by φ P (s, (r, r ′ , θ)) = ( √ sr, √ sr ′ , θ).Now, using Theorem 3.5, we can further reduce again the system (second step) with this last scaling symmetry.We obtain: • The reduced space: Consider the diffeomorphim which transforms the generator ∆ P of the R + -action on R + × R + × S 1 into the vector field 1 2 ρ∂ ρ with (ρ, ρ ′ , σ) local coordinates on R + × (R + × S 1 ).Thus, the space of orbits of the reduced R + -action may be identified with and, under this identification, the canonical projection is The associated line bundle is trivial and therefore, we have a Jacobi bracket on the space of functions on K.In the sequel we will describe this structure.
The expression of the reduced Poisson structure on P in terms of the new local coordinates (ρ, ρ ′ , σ) is (see ( 33)) ( 35) Note that L 1 2 ρ∂ρ Π P = −Π P .Since the homogenous functions with respect to the vector field 1  2 ρ∂ ρ are of the form ρ 2 h, with h ∈ C ∞ (R + × S 1 ), then we have that ).This implies that the Jacobi bracket {•, •} K on the space of functions on K satisfies As a consequence (see (35)), Therefore, the corresponding Jacobi structure • The reduced Hamiltonian function: The Hamiltonian function H S 1 (see (34)), in terms of the local coordinates (ρ, ρ ′ , σ), is • The reduced dynamics: The Hamiltonian vector field induced by the previous Jacobi structure and the function h which is the p P -projection of X lift action, i.e. the free and proper action given by (38) (T * Φ) g (α q ) = (T Φg (q) Φ g −1 ) * (α q ), ∀g ∈ G and ∀α q ∈ T * q Q − 0 q .It is well-known that (T * Φ) g is a symplectomorphim with respect to the standard symplectic structure ω Q on

the action given by (8).
A direct computation, using ( 38) and ( 39), shows that the fiberwise-linear function The symplectic action T * Φ is fiberwise linear.So, Thus, the previous comments imply that the actions T * Φ and φ are compatible and the conditions of Theorem 4.3 hold.Now, we will reduce the Hamiltonian symplectic system (T * Q − 0 Q , ω Q , Y ℓ ), first by T * Φ and then by the scaling symmetry φ.The objets obtained after the G-reduction are: • The reduced space: The restriction of the canonical projection τ * Q : • The reduced dynamics: The Hamiltonian vector field X ωQ Y ℓ is ℘-projectable and its projection is just • The scaling symmetry on the reduced space: The scaling symmetry φ : Now, we will apply the second reduction step to the Poisson Hamiltonian system (P = (T * Q−0 Q )/G, Π P , (Y ℓ ) G ). with respect to the scaling symmetry φ G : (R − {0}) × P → P. The reduced objects in this second reduction are: • The reduced space: In this case, the reduced space is the projective bundle The particular case of a Lie group.In what follows, we will show the previous reduction process in the particular case when the initial manifold Q is a Lie group G.In such a case, one may use the left trivialization of the cotangent bundle T * G in order to identify T * G with the product manifold G × g * , where (g, [•, •] g ) is the Lie algebra of G, in such a way that the canonical projection τ * G : T * G → G is just the first projection p 1 : G × g * → G.The left action Φ : G × G → G on G is the one defined by the group operation of G.We take the left invariant vector field Y = ← − ξ on G induced by an element ξ of g.In the first reduction with the cotangent lift of Φ, the reduced space is (T * G − 0 G )/G ∼ = g * − {0} and the reduced function induced by Y is the restriction to g * − {0} of the linear map ξ ℓ associated with ξ ∈ g, i.e.
The Kirillov bracket on the projective space Pg * is characterized by This structure on the line bundle L → Pg * may be considered as the Kirillov version of the Lie-Poisson structure on g * and for this reason we will use the terminology the Lie-Kirillov structure on Pg * .
The reduced dynamics is determined by the p-projection of the Lie-Poisson Hamiltonian vector field associated with the linear function is just the vector field X h ξ ∈ X(Pg * ), which is locally characterized by (22).

Reduction of symplectic Hamiltonian systems using first the scaling symmetry and then the standard symmetries
As in the previous section, we have a symplectic Hamiltonian system (S, ω, H) with a scaling symmetry φ S : R × × S → S and a symplectic G-symmetry Φ S : G × S → S which are compatible.In what follows we describe the reduction process of the system (S, ω, H) in two steps, but in the following order: the first reduction is obtained by the scaling symmetry and the second step is done using the standard symmetry.
First of all, we will show a reduction process for Kirillov structures in the presence of a standard symmetry.We suppose that ( is a representation of a Lie group G on the vector bundle π L : L → K.This means that (Φ L g , Φ K g ) is a vector bundle isomorphism for every g ∈ G. So, we have a dual representation (Φ We recall that a section h : On the other hand, since the principal bundle associated with π L is the restriction p L−0L : In what follows, we suppose that the orbit space K/G of the action Φ K of G on K is a smooth quotient manifold.As a consequence, the orbit space L/G is a real line bundle over K/G whose fibers are isomorphic to the fibers of π L : L → K.

Denote by 0 L/G the zero section of the line bundle π
Moreover, the principal actions φ L−0L and φ (L−0L)/G of R × on L − 0 L and (L − 0 L )/G, respectively, are related by ( 42) On the other hand, the dual vector bundle π * L/G : (L/G) * → K/G is isomorphic to the line bundle π L * /G : L * /G → K/G deduced from the G-equivariant dual vector bundle π L * : L * → K of π L for the pair of actions (Φ L * , Φ K ).The following diagram summarizes the previous comments Now we can prove the following general result that will be used in the following.
) be a Kirillov structure on the real line bundle π L : L → K. Suppose that (Φ L , Φ K ) is a compatible representation of G on L. Then: (1) There is a one-to-one correspondence between G-equivariant sections h : Proof.From the general theory of representations of Lie groups, we have that there is a one-to-one correspondence between G-equivariant sections h :

Now, by hypothesis, the sections [(f • ℘
So, we have proved that the vector field Therefore, [•, h G 2 ] L * /G is a derivation and its symbol is just the ℘ K -projection of the symbol of [•, h 2 ] L * .This finishes the proof of the theorem.
The following diagram summarizes this reduction process Remark 5.3.When the real line bundle π L : L → K is trivial, the previous theorem is just the reduction process of Jacobi manifolds given in [42].
5.2.The first step: Reduction by a scaling symmetry.Now, we start with the scaling reduction process of the symplectic Hamiltonian system (S, ω, H).In this case we have (see Section 3): • The reduced space C = S/R × admits a contact distribution D.
• The principal bundle p S : , where {•, •} S is the Poisson bracket associated with the symplectic structure on S.
• The Hamiltonian vector field X ω H ∈ X(S) of H with respect to the symplectic structure ω is p S -projectable on C and its projection is the symbol X , for all g ∈ G and s ∈ R × , implies that Φ S g : S → S is R × -equivariant and, therefore, it induces a principal action Φ with ∆ the infinitesimal generator of the scaling symmetry φ S .Using this relation and that (Φ S g ) * ω = ω, we conclude the G-invariance of the 1-form λ = −i ∆ ω, i.e. (46) ( where T Φ S : G × T S → T S is the tangent lift of the action of Φ S .In other words, To do so, we will use Theorem 5.2 on the reduction of Kirillov structures. Theorem 5.4.Let (S, ω, H) be a symplectic Hamiltonian system with a scaling symmetry φ S : R × × S → S, G a Lie group and Φ S : G × S → S a symplectic G-symmetry which is compatible with φ S .Then: (1) If (C = S/R × , D) is the contact manifold induced by the scaling symmetry φ S , then we have a representation Moreover, there is a one-to-one correspondence between the G-equivariant sections h : C → (D o ) * of the dual vector bundle of π of the sections of the dual vector bundle π 70), (71) (see Appendix A) and the commutation of the actions Φ S and φ S , we deduce that h : C → (D o ) * is a G-equivariant section if and only if the corresponding homogeneous function H h : S → R is invariant with respect to the action Φ S .
then H h1 and H h2 are G-invariant with respect to Φ S and, since the action Φ S is symplectic, we have that the function {H h1 , H h2 } S is G-invariant.Therefore, Now, applying Theorem 5.2, we deduce the result.
The following diagram shows both reduction processes together R (S, ω, X ω H ) H 8 8 q q q q q q q q q q q q pS / / (C = S/R × , D, X Now we illustrate the reduction processes using the two examples considered above. Example 5.5 (Continuing Example 4.4: The 2d harmonic oscillator reduced first by a scaling and then by a standard symmetry).We consider again the example of a 2-dimensional harmonic oscillator (see Examples 3.2 and 4.4).In Example 4.4 we have shown how to apply the reduction process by first using the standard symmetry and then the scaling symmetry.Now, we take the reverse order.
We recall that in this example we have: (1) A standard rotational S 1 -symmetry, with infinitesimal generator ξ S = x∂ y − y∂ x + p x ∂ py − p y ∂ px where (x, y, p x , p y ) are coordinates on S = T * (R 2 − {(0, 0)}).Using the identification where (r, θ, r ′ , θ ′ ) are polar coordinates on As seen in Example 3.2, by applying first the scaling R + -symmetry, we obtain: • The reduced space: , with the quotient map The Jacobi structure on C = R + × S 1 × S 1 is given by ( 16).• The reduced Hamiltonian function: The reduced Hamiltonian function is given by • The reduced dynamics: It is given by the vector field on R + × S 1 × S 1 obtained by the p-projection which is just the contact Hamiltonian vector field X {•,•}C with respect to the Jacobi structure on C = R + × S 1 × S 1 described in (16).
• The standard symmetry on the reduced space: We may induce an S 1 -action on the reduced space R + × S 1 × S 1 whose infinitesimal generator is Now, we apply the second step of the reduction process using this last symmetry, obtaining the reduction of the Kirillov structure by this standard symmetry.More precisely, the reduction of the Jacobi structure, because in this case the Kirillov line bundle is trivial.
• The reduced space: We consider the diffeomorphism Therefore, the quotient space (R + × S 1 × S 1 )/S 1 may be identified with In this case, the line bundle associated with ℘ K is trivial and we obtained a Jacobi structure.From ( 16) we deduce that the Jacobi structure on with (ρ ′ , σ) polar coordinates on R + × S 1 .Note that this Jacobi structure is just the one given in (36).
• The reduced Hamiltonian function: In this case, the reduced Hamiltonian is • The reduced dynamics: The reduced vector field is the ℘ K -projection which coincides precisely with the results obtained in Example 4.4, using the reverse reduction process (see (37)).
Example 5.6.Continuing Example 4.5: The linear Hamiltonian system reduced first by a scaling and then by a standard symmetry.We consider again the example of a free and proper action Φ : G × Q → Q of a Lie group G on a manifold Q with a G-invariant vector field Y ∈ X(Q).Then, we have two symmetries on (8).
In Example 4.5 we have shown how to apply the reduction process by first using the standard symmetry and then the scaling symmetry.Now, we take the reverse order.
As seen in Example 3.3, by using first the scaling symmetry, we obtain the following reduced objects: • The reduced space: It is the projective cotangent bundle P(T * Q).Let D be the contact distribution on P(T * Q) such that p : The Kirillov bracket on the sections of for all X, Z ∈ X(Q).• The reduced Hamiltonian section: It is defined locally by (19).
• The reduced dynamics: The Hamiltonian vector field  • The reduced vector field after this reduction is (Y, X h ξ ) ∈ X(G) × X(Pg * ), such that (49) Now, if we perform the second reduction step associated with the induced G-action the corresponding reduced elements are: • The reduced space is the projective space Pg * .
• The line vector bundle π L : L → Pg * is given by • The reduced section of π L * : L * → Pg * is just • The final reduced dynamics is the vector field X h ξ on Pg * described in (49), which is the symbol of [•, h G ξ ] L * and whose local expression is (22).So, also in this case, similarly to the two previous examples (see Examples 4.4,4.5,5.5 and 5.6), both reduction processes give rise to the same reduced dynamics.This fact motivates further analysis on the equivalence of the two reduction processes, which will be addressed in full generality in the following section.

The equivalence of the two reduction processes
Finally, we will prove that both processes considered in Sections 4 and 5 are equivalent.Let (S, ω, H) be a symplectic Hamiltonian system with a scaling symmetry φ S : R × × S → S and a symplectic G-symmetry Φ S : G × S → S which are compatible, G being a Lie group.Theorem 6.1.Under the previous conditions we have that: (1) There exists a real line bundle isomorphism (Ψ, ψ) between the line bundles π L : L → (S/G)/R × and isomorphic.In fact, we have that given in Theorem 4.3 and Theorem 5.4 respectively, are ψ-related, i.e. the following diagram is commutative The diffeomorphism ψ is just (52) ψ that is, We remark that this map is a diffeomorphism from the equality Φ S g •φ S s = φ S s •Φ S g .Moreover, the diffeomorphism Ψ is characterized in this diagram (53) S × R Here p P×R is the quotient map deduced from the action and p S×R the quotient map deduced from the action (2) From (71) in Appendix A, we have for x ∈ S and t ∈ R.
On the other hand, using (52), the diagram (53) and again (71) in Appendix A, we obtain On the other hand, using b) in Theorem 3.5, ( 29) and ( 50), we have Therefore, we have (51).
(4) We consider the section On the other hand, using ( 50) and ( 51), we obtain Replacing these relations in (6), we have that However, we know that Comparing ( 54) and (55), we conclude (4).
Both reduction processes and the corresponding equivalence between them are summarized in the following diagram R R

Reconstruction process for scaling symmetries
In this section we will study the inverse process of reduction: the reconstruction process.First, we shall introduce the general involved ideas, for arbitrary dynamical systems and Lie groups, and then we shall concentrate on the case of symplectic Hamiltonian systems with scaling symmetries.7.1.The general context.Let M be a manifold, X ∈ X (M ) a vector field on M and G a Lie group acting on M by an action φ M : G × M → M such that X is G-invariant.Assume that φ M defines a principal fiber bundle p M : M → M/G.In such a case, the G-invariance of X ensures that there exists a vector field X G ∈ X (M/G) such that X G • p M = T p M • X.The question is: how can we get the integral curves of X from those of X G ?To do that, we can proceed as follows.If we want the integral curve Γ : (−ǫ, ǫ) → M of X such that Γ (0) = x 0 , then: (1) consider the integral curve γ : (−ǫ, ǫ) → M/G of X G such that γ (0) = p M (x 0 ); (2) fix a principal connection A : T M → g for p M (where g is the Lie algebra of G) and fix a curve ϕ : (in other words, t → ϕ(t) is the horizontal lift of the curve γ by the principal connection A); (3) and find the curve g : From now on, we shall take ǫ small enough in order to fulfill above conditions.Then, proceeding as in [1] (see pages 304-305), one may prove that Γ (t) = φ M (g (t) , ϕ (t)) is the curve we are looking for.The above three-step procedure is usually known as reconstruction.The steps 2 and 3 are known as the reconstruction problem (see, for example [37]).
Clearly, such a procedure can be used for the standard as well as for the scaling symmetries.In the following, we shall focus on the latter, since the reconstruction process for scaling symmetries, as far as the authors know, has not been studied in the literature so far.• and we can ensure that the Hamiltonian vector field X ω H ∈ X(S) of H projects onto the symbol X Recall that h H : C → (D o ) * denotes the section of Γ((D o ) * ) related to the homogeneous function H. So, we are in the situation of the previous subsection, with . We shall apply the reconstruction procedure described above in this particular context.

Existence of a flat connection.
There is a case in which solving the reconstruction problem is especially simple (as we will show later).This case is when there is a non-vanishing homogeneous function F : S → R × .This kind of functions are called scaling functions [12].
In such a case the map (F, p S ) : S → R × × C is a diffeomorphism and defines a trivialization for p S .Its inverse is given by Conversely, if p S : S → C is trivial, i.e. S ∼ = R × × C and p S is the second projection, then the function F : S ∼ = R × × C → R given by F (s, x) = s is a non-vanishing homogenous function, i.e. a scaling function.
Therefore, the existence of a scaling function F on S is equivalent with the trivialization of the principal bundle p S : S → C.This fact guarantees the local existence of this kind of functions F (see [12]).
Moreover, if ∆ is the infinitesimal generator of φ, since On the other hand, the 1-form η := σ * (λ) is a global generator of D o with λ = −i ∆ ω, which makes π D o trivial.In fact, using (58), we have that σ Since λ = −i ∆ ω, then T * s φ x (λ(φ(x, s)) = 0, for all (s, x) ∈ R × × S. Thus, from the homogeneity of λ, we have that In conclusion, we deduce that This implies that D = η o , and η is a contact 1-form on C.
The one-to-one correspondence between homogeneous functions H : S → R on S and functions h H : C → R on C (sections of the trivial line bundle π (D o ) * ) is defined by the relation Note that the function on C associated with F is just the constant function 1.
The Jacobi bracket of two functions h 1 , h 2 on C defined by the contact 1-form η is given by The relation between the Hamiltonian vector field X ω H of H with respect ω and the Hamiltonian vector field X η hH of h H with respect to the contact structure η is T p S • X ω H = X η hH • p S .Remark 7.1.Since above equation is actually true for any homogeneous function H, for H = F we have that where R is the Reeb vector field of η.
Thus, the step 2 is complete.
In order to find the curve g (t), let us calculate A (X ω H (ϕ (t))).Using the decomposition T S = ∆ ⊕ dF o , we have that Thus, we have found, up to quadratures, the trajectories Γ (t) of X ω H from the trajectories γ (t) of X η hH .
Remark 7.3.According to the local existence of scaling functions, if there is not a (global) scaling function for φ S , then we can proceed as above around every point x ∈ S, just replacing S by an appropriate coordinate neighborhood U of x 0 .In particular, we can obtain the result of Remark 7.2 along the open submanifold of S where H = 0.
To end this section, suppose that, instead of a symplectic Hamiltonian system, we have a Poisson Hamiltonian system (P, Π, H) with scaling symmetry φ P : R × × P → P such that p P : P → K = P/R × is a principal bundle.Assume that F : P → R × is a scaling function for φ P .Then, as we saw above, the related line bundle π L : L → K is trivial (via a global section as that given by (58)), so the sections of π L * can be identified with the functions h : K → R, which in turn are in bijection with the homogeneous functions H : P → R through the equation h H • p P = 1 F H. Also, the related Kirillov bracket [•, •] L * can be identified with the Jacobi bracket {•, •} K given by {h H1 , h H2 } K • p P = − 1 F {H 1 , H 2 } P .
Moreover, following the same calculations made along this section for the symplectic case, given x 0 ∈ P , we can construct the trajectory Γ (t) of X {•,•} P H such that Γ (0) = x 0 , in terms of the trajectory γ (t) of X which is a scaling function.
Example 7.5.The projective cotangent Hamiltonian system deduced from a standard linear Hamiltonian system.We consider Example 3.3 with Y ∈ X(Q) a vector field on the manifold Q of dimension n.Let U i0 be the open subset of T * Q − 0 Q given by U i0 = {(q 1 , . . .q n , p 1 , . . .p n ) ∈ T * Q − 0 Q /p i0 = 0}, with (q i , p i ) local coordinates on T * Q.The local expressions of the linear function Y ℓ and of the corresponding Hamiltonian vector field X ωQ Y ℓ are Y ℓ (q, p) = Y i (q)p i and Note that Y ℓ is a scaling function if and only if Y is a vector field without zeros.In any case, we have a scaling function on U i0 given by F : U i0 → R, F (q i , p i ) = p i0 After the reduction process of the Hamiltonian symplectic system (T * Q − 0 Q , ω Q , H) by the scaling symmetry (8), we have that the local expressions of the reduced elements are: • The local expression of the projective bundle p : T * Q − 0 Q → P(T * Q) on U i0 : p(q 1 , . . .q n , p 1 , . . .p n ) = (q 1 , . . .q n , p 1 p i0 , . . ., p i0−1 p i0 , p i0+1 p i0 , . . ., p n p i0 ).
The local expression of the line bundle π D o : D o → P(T * Q) on U i0 is π D o (q, p i , t) = (q, p i ).
4).•The reduced Hamiltonian function: Denote by π L : L → P(T * Q/G) the line bundle associated withp P : (T * Q − 0 Q )/G → P(T * Q/G).The section of the dual bundle π L * : L * → P(T * Q/G) induced by the homogeneous function (Y ℓ ) G ∈ C ∞ ((T * Q − 0 Q )/G)is the reduced Hamiltonian function.• The reduced dynamics: The Hamiltonian vector field X {•,•}P (Y ℓ ) G is p P -projectable and it determines the final reduced dynamics.

5. 1 .
Reduction of Kirillov structures by standard symmetries.Let π L : L → K be a real line vector bundle with a Kirillov bracket[•, •] L * : Γ(L * ) × Γ(L * ) → Γ(L * )on the space of the sections Γ(L * ) of the dual vector bundle π L * : L * → K of π L .Denote by 0 L the zero section of π L and by φ L−0L : R × × (L − 0 L ) → (L − 0 L ) the R × -action associated with the principal bundle p L−0L : (L − 0 L ) → K whose line bundle is π L (see Appendix A).
where D o is the annihilator of D and 0 C is its zero section.Therefore, the associated real line bundle, under this isomorphism, is π D o : D o → C.Moreover, there is a one-to-one correspondence between the sections h : C → (D o ) * of the dual vector bundle of π D o and the homogeneous functions H h : S → R on the symplectic manifold S.• On the space Γ((D o ) * ) of the sections of the dual vector bundle of π D o , we have a Kirillov bracket

5 . 3 .
T Φ C g (D) = D.This implies that the cotangent lift T * Φ C of the action Φ C preserves the annihilator D o of the contact distribution.Therefore, we have a representation (Φ D o := (T * Φ C ) |D o , Φ C ) of G on the real line bundle π D o : D o → C. The second step: Reduction by standard symmetries.Now, we apply the second reduction process with the representation (Φ D o and its projection is the symbol of the derivation [•, h Y ℓ ] (D o ) * .• The standard symmetry on the reduced space: The action is defined by G × P(T * Q) → P(T * Q), (g, p(α)) → p((T * Φ) g (α)).Now, we can consider the second step of the reduction process.The standard symmetry on the reduced space satisfies the conditions of Theorem 5.2, and therefore, we have • The reduced space: In this case, the reduced space is the quotient space P(T * Q)/G.Moreover, the projection p :D o − 0 → P(T * Q) is G-invariant and it induces a reduced projection p G :(D o − 0)/G → P(T * Q)/G.The real line bundle π D o /G :L := D o /G → K := P(T * Q)/G is deduced from the G-equivariant line bundle π D o .On the space of sections of the dual of this real bundle we have a Kirillov structure [ vector fields on Q.•The reduced Hamiltonian section: The section hY ℓ of π (D o ) * : (D o ) * → P(T * Q) is G-invariant and therefore it induces a section h G Y ℓ : P(T * Q)/G → (D o ) * /G.• The reduced dynamics: The vector field p * (X ωQ Y ℓ ) is G-invariant.Thus, it induces a vector field on P(T * Q)/G, which is just the symbol of [•, h G Y ℓ ] L * .The particular case of a Lie group.When Q = G is a Lie group, for the first reduction step with the scaling symmetry, we have (see Example 3.3):• The reduced space is G × Pg * .•The contact structure is the distribution on G × Pg * given byD (g,p(µ)) = (T g L g − 1 ) * (µ) o × T p(µ) (Pg * )for all g ∈ G and µ ∈ g * − {0}.• The fiber of the real line bundle πD o : D o → G × Pg * at (g, p(µ)) ∈ G × Pg * is just D o (g,p(µ)) = (T g L g − 1 ) * (µ) .• The reduced Hamiltonian section of π (D o ) * : (D o ) * → G × Pg * induced by the function Y ℓ is characterized by h ξ (g, p(µ))((T g L g −1 ) * (µ)) = µ(ξ), with g ∈ G, µ ∈ g * − {0}, ξ = Y (e)and p : g * − {0} → Pg * the corresponding quotient map determined by the scaling symmetry on g * − {0}.

7. 2 .
Application to scaling symmetries and symplectic Hamiltonian systems.Now, as in Section 3, let us suppose that we have a scaling symmetry φ : R × × S → S on a symplectic Hamiltonian system (S, ω, H), with infinitesimal generator ∆.Then, assuming that p S : S → C = S/R × is a principal bundle (see the first part of Section 3),•we have a contact distribution D on C and a related real line bundle π D o : D o → C with a Kirillov structure [•, •] (D o ) * , for all s ∈ R × and x ∈ S, and we have a global section σ : C → S of p S which takes the values (58) σ (y) = (F, p S ) −1 (1, y) , ∀y ∈ C.
.3.Examples.In this subsection we will apply the previous reduction processes to Examples 3.2 and 3.3.Example 4.4.Continuing Example 3.2: The 2d harmonic oscillator reduced first by a standard and then by a scaling symmetry.In this case we have: (1) A standard rotational S 1 -symmetry, with infinitesimal generator ξ S = x∂ y − y∂ x + p x ∂ py − p y ∂ px , where (x, y, p x , p y ) are coordinates on Example 4.5 (Continuing Example 3.3: The linear Hamiltonian system reduced first by a standard and then by a scaling symmetry).Let Φ : G×Q → Q be a free and proper action of a Lie group G on a manifold Q. Denote by 0 Q the zero section of the cotangent bundle τ * Q a scaling symmetry φ G : (R − {0}) × P → P for the reduced Poisson Hamiltonian system (P, {•, •} P , (Y ℓ ) G ) which is given by