On the Geometrical Gyro-Kinetic Theory

Considering a Hamiltonian Dynamical System describing the motion of charged particle in a Tokamak or a Stellarator, we build a change of coordinates to reduce its dimension. This change of coordinates is in fact an intricate succession of mappings that are built using Hyperbolic Partial Differential Equations, Differential Geometry, Hamiltonian Dynamical System Theory and Symplectic Geometry, Lie Transforms and a new tool which is here introduced : Partial Lie Sums.


7.
For an open subset U Ă R p , we denote by A pU q the space of real analytic functions on U .
8. For a formal power series S, we denote by Σ S its set of convergence.
9. b n pm 0 , R 0 q stands for the open euclidian ball of radius R 0 and of center m 0 in R n . 10. b #`m 0 , R m 0˘s tands for !

Introduction
At the end of the 70', Littlejohn [22,23,24] shed new light on what is called the Guiding Center Approximation. His approach incorporated high level mathematical concepts from Hamiltonian Mechanics, Differential Geometry and Symplectic Geometry into a physical affordable theory in order to clarify what has been done for years in the domain (see Kruskal [21], Gardner [10], Northrop [25], Northrop & Rome [26]). This theory is a nice success. It has been beeing widely used by physicists to deduce related models (Finite Larmor Radius Approximation, Drift-Kinetic Model, Quasi-Neutral Gyro-Kinetic Model, etc., see for instance Brizard [1], Dubin et al. [3], Frieman & Chen [8], Hahm [15], Hahm, Lee & Brizard [17], Parra & Catto [28,29,30]) making up the Gyro-Kinetic Approximation Theory, which is the basis of all kinetic codes used to simulate Plasma Turbulence emergence and evolution in Tokamaks and Stellarators (see for instance Brizard [1], Quin et al [31,32], Kawamura & Fukuyama [20], Hahm [16], Hahm, Wang & Madsen [18], Grandgirard et al. [13,14], and the review of Garbet et al. [9]). Yet, the resulting Geometrical Gyro-Kinetic Approximation Theory remains a physical theory which is formal from the mathematical point of view and not directly accessible for mathematicians. The present paper is a first step towards providing a mathematical affordable theory, particularly for the analysis, the applied mathematics and computer sciences communities.
The purpose of this paper is to provide a mathematical framework for the formal Guiding-Center reduction introduced in Littlejohn [22]. The domain of application of this theory is that of a charged particle under the action of a strong magnetic field. Hence we will consider 3 the following dynamical system : where X " pX 1 , X 2 q stands for the position, V " pV 1 , V 2 q stands for the velocity, V K " pV 2 ,´V 1 q, x 0 and v 0 stand for the initial position and velocity, and ε is a small parameter. We notice that equations (1.1)-(1.2) can be obtained from the six dimensional system by taking a magnetic field in the x 3 -direction that only depends on x 1 and x 2 . When the magnetic field is constant, the trajectory associated with (1.1)-(1.2) is a circle of center c 0 " x 0`ε v 0 and of radius ε |v 0 |. Otherwise, the dynamical system (1.1)-(1.2) can be viewed as a perturbation of the system obtained when the magnetic field is constant. Hence, in the general case of a magnetic field depending on position, the evolution of a given particle's position is a combination of two disparate in time motions: a slow evolution of what is the center of the circle in the case when B is constant, usually called the Guiding Center, and a fast rotation with a small radius about it. The Guiding-Center reduction consists in replacing the trajectory of the particle by the trajectory of a quantity close to the guiding-center and free of fast oscillations. This purpose can easily be translated within a geometric formalism. In any system of coordinates on a manifold M, a Hamiltonian dynamical system whose solution is R " Rpt; r 0 q can be written in the following form BR Bt " PpRq∇ r HpRq, Rp0, r 0 q " r 0 , where Pprq is a matrix called the matrix of the Poisson Bracket (or Poisson Matrix in short), and Hprq is called the Hamiltonian function. The Poisson Matrix is a skew-symmetric matrix satisfying the Jacobi identity and the Hamiltonian function is a smooth function (see Appendix A). It is obvious to show that dynamical system (1.1)-(1.2) is Hamiltonian and to find its related Poisson MatrixP ε px, vq and Hamiltonian functionH ε px, vq (see Section 2.1). Within this geometrical framework, the goal of the Guiding-Center reduction is to make a succession of changes of coordinates in order to satisfy the assumptions of the following theorem.
Theorem 1.1. If, in a given coordinate system r " pr 1 , r 2 , r 3 , r 4 q, the Poisson Matrix has the following form: Pprq "¨M prq 0 0 Consequently, the time-evolution of the two first components R 1 , R 2 is independent of the penultimate component R 3 ; and, the last component R 4 of the trajectory is not time-evolving, i.e.

BR 4
Bt " 0. (1.7) Theorem 1.1 is the Key Result that brings the understanding of the Guiding-Center reduction: the Guiding-Center reduction consists in writing dynamical system (1.1)-(1.2) within a system of coordinates, called the Guiding-Center Coordinate System, that satisfies the assumptions of Theorem 1.1 and which is close to the Historic Guiding-Center Coordinate System, usually defined by: where v " |v| and where θ is the angle between the x 1 -axis and the gyro-radius vector ρ ε px, vq "´ε Bpxq v K measured in a clockwise sense. Once this done, if we are just interested in the motion of the particle in the physical space, i.e. just in the evolution of the two first components, solving the dynamical system in the new system of coordinates, reduces to find a trajectory in R 2 , in place of a trajectory in R 4 when it is solved in the original system of coordinates.
In [22], Littlejohn proposed a construction of the Guiding-Center Coordinates based on formal series expansion in power of ε. This approach cannot be made mathematically rigorous because no argument can insure the validity of the series expansion. In the present paper we adopt a different strategy. We will derive for each positive integer N a coordinate system, the so-called Guiding-Center Coordinates of order N , whose expansion in power of ε, up to any order N , coincides with the Guiding-Center coordinates given in [22]. Moreover, for each integer N we will construct a Hamiltonian dynamical system satisfying Theorem 1.1 and approximating uniformly in time, with accuracy in proportion to ε N´1 , the Hamiltonian dynamical system (1.1)-(1.2) written within the Guiding-Center Coordinates of order N .
The Guiding-Center reduction consists essentially in a succession of three change of coordinates: a polar in velocity change of coordinates px, vq Þ Ñ px, θ, vq with θ and v defined above, a second change of coordinates called the Darboux change of coordinates, and a last change of coordinates called the Lie change of coordinates. The objective of the first change of coordinates is to concentrate the fast oscillations on the θ variable. The second one, consists in finding a coordinate system in which the Poisson Matrix has the required form to apply Theorem 1.1, and eventually the last change of coordinates (which is in fact the succession of N changes of coordinates) consists in removing the oscillations from the Hamiltonian function while keeping the same expression of the Poisson Matrix.
All along this paper we will assume that the magnetic field B is analytic, that all its derivatives are bounded, and that B is nowhere close to 0, i.e. that inf B ą 1.
The three main results of this paper are the following. The first one concerns the Darboux change of coordinates.
There exists a C 8 -diffeomorphism Υ ε : px, θ, vq Þ Ñ py, θ, kq one to one from R 2ˆRˆp 0,`8q onto itself, smooth with respect to ε, such that the Poisson Matrix expressed in the py, θ, kq coordinate system reads: Moreover, the reciprocal map κ ε " Υ´1 ε is smooth with respect to ε P R`, and for any positive real numbers a D and b D (with a D ă b D ) and for any t P r0,`8q, the trajectory associated with (1.1)-(1.2), with initial condition px 0 , v 0 q P R 2ˆC pa D , b D q (see Notation 11) and expressed in the Darboux coordinates, belongs to R 3ˆ" a 2 Subsections 3.4, 3.5, 3.6 and 3.7 constitute the proof of Theorem 1.2. Theorem 1.3. For each positive integer N , for each compact set K L , and for each positive real numbers c L and d L (with c L ă d L ), there exists a diffeomorphism χ N ε : py, θ, kq Þ Ñ pz, γ, jq defined on K LˆRˆp c L , d L q and a positive real number η K L such that, for any ε P r0, η K L s, the expansion in power of ε of the Hamiltonian functionĤ ε of system (1.1)-(1.2) in the pz, γ, jq coordinates does not depend to the oscillation variable γ up to order N , i.e. 13) and such that the Poisson Matrix expressed in the pz, γ, jq coordinate system reads: 14) where ρ N H and ρ N P are in C 8 # pr0, η K L sˆK LˆRˆr c L , d L sq. The proof of Theorem 1.3 is given in Subsection 4.7.
Remark 1.4. Theorem 1.3 is consistent with Theorem 1.2. Indeed, in Subsection 3.9 we will show that for any T P r0,`8q, for any compact set K C , and for any positive real numbers c C and d C (with c C ă d C ), there exists a positive real number η, a compact set K L , and positive real numbers c L and d L (with c L ă d L ) such that for any t P r0, T s and for any ε P r0, ηs the trajectory associated with (1.1)-(1.2), with initial condition px 0 , v 0 q P K CˆC pa C , b C q (see Notation 11) and expressed in the Darboux coordinates, belongs to K LˆRˆr a L , b L s. Theorem 1.5. With the same notations as in Theorem 1.3, we consider the functionĤ N ε defined by:Ĥ N ε pz, jq "Ĥ 0 pz, jq`εĤ 1 pz, jq`. . .`ε NĤ N pz, jq , (1.15) whereĤ 0 , . . . ,Ĥ N are the N first terms in expansion (1.13) ofĤ ε , and we denote by pZ, Γ, J q " pZ, Γ, J qpt; z 0 , γ 0 , j 0 q the trajectory of Hamiltonian system (1.1)-(1.2), expressed in the pz, γ, jq coordinate system, associated with initial condition z 0 , γ 0 , j 0 . Let be the Hamiltonian dynamical system associated with the Hamiltonian functionĤ N ε and with the Poisson Matrix P ε defined by (1.12). Then, this system satisfies the assumptions of Theorem 1.1. Moreover, for any T P r0,`8q, for any compact set K C , and for any positive real numbers c C and d C (with c C ă d C ), there exists a real number η K C and a constant C C , independant of ε, such that for any t P r0, T s and for any ε P r0, where U C is the range of K CˆC pc C , d C q in the Guiding-Center coordinates of order N i.e. by diffeomorphism χ N ε . The proof of Theorem 1.5 is led in Subsection 4.8.
The paper is organized as follows. In Section 2 we briefly recall the main steps of the Guiding-Center reduction and we give a proof of Theorem 1.1. Then, Section 3 is devoted to the construction of the Darboux change of coordinates. Especially, we will introduce an intermediary PDE from which the Darboux coordinates can be deduced. We will also perform a detailed analysis of the regularity of the change of coordinates and its inverse, including the regularity with respect to the small parameter ε, and we will give the expansions with respect to ε of the change of coordinates, of its inverse, and of the Hamiltonian function. In Section 4, we introduce a partial Lie transform method leading to the Guiding-Center coordinate system of order N . Eventually, in Sections 4.7 and 4.8 we will prove Theorems 1.3 and 1.5.
2 Schematic description of the Guiding-Center reduction

Panorama
A schematic description of the Guiding-Center change of coordinates is summarized in Figure  2.1. The three main steps of the reduction was already discussed in the introduction. They  are symbolized by arrows 3, 4, and 5. The first step consists in finding an adequate symplectic structure from which the expressions of the Poisson Matrix and the Hamiltonian function are deduced. To achieve this goal we will introduce the canonical coordinates defined by: is the dimensionless electromagnetic Lagrangian and A is the potential vector. Then, the Symplectic Two-Form Ω ε that is considered is the unique Two-Form whose expression in the Canonical Coordinate chart is given by Consequently, the Poisson matrix is given by: whereK ε is the matrix associated withω ε . Eventually it is obvious to show that dynamical system (1.1)-(1.2) is Hamiltonian with Hamiltonian functionH ε pq, pq " 1 2ˇp´1 ε A pqqˇˇ2 and Poisson MatrixP ε .
Using the usual change of coordinates rules for the Poisson Matrix and the Hamiltonian function (see Appendix A) we obtain the following expressions of the Hamiltonian function and of the Poisson Matrix, in the Cartesian Coordinates: Hence, if the Hamiltonian function does not depend on the penultimate variable, then, the last component R 4 of the trajectory is not time-evolving. Now, introducing the Poisson Bracket of two functions f " f prq and g " g prq defined by tf, gu r prq " r∇ r f prqs T P prq ∇ r g prq , where Pprq is the Poisson Matrix, we have where r i is the i-th coordinate function r Þ Ñ r i and a direct computation leads to ttr 1 , r 2 u r , r 3 u r prq "´P 3,4 BP 1,2 Br 4 prq and ttr 1 , r 2 u r , r 4 u r prq " P 3,4 BP 1,2 Br 3 prq . (2.10) Using the Jacobi identity saying that for any regular function f, g, h, ttf, gu r , hu r`t th, f u r , gu r`t tg, hu r , f u r " 0, (2.11) and the facts that P 3,1 " P 2,3 " P 4,1 " P 2,4 " 0, we obtain ttr 1 , r 2 u r , r 3 u r "´ttr 3 , r 1 u r , r 2 u r´t tr 2 , r 3 u r , r 1 u r " 0, (2.12) ttr 1 , r 2 u r , r 4 u r "´ttr 4 , r 1 u r , r 2 u r´t tr 2 , r 4 u r , r 1 u r " 0.

Objectives
At this stage, the three first steps of the reduction are already done. The fourth step (see Figure 2.1) on the way to build the Guiding-Center Approximation is the application of the mathematical algorithm, so called the Darboux Algorithm, to build a global Coordinate System py 1 , y 2 , θ, kq close to the Historic Guiding-Center Coordinate System (1.8)-(1.11), and in which the Poisson Matrix has the required form (1.4) to apply the Key Result (Theorem 1.1). In order to manage the small parameter ε, we will build the Coordinate System py 1 , y 2 , θ, kq in order to haveP ε py, θ, kq with the following form: P ε py, θ, kq "¨M ε pyq 0 0 Using the usual change of coordinates rule for the Poisson Matrix, finding this coordinate system remains to find a diffeomorphism Υ px, θ, vq " pΥ 1 px, θ, vq , Υ 2 px, θ, vq , Υ 3 px, θ, vq , Υ 4 px, θ, vqq , (3.2) whose components satisfy the following non-linear hyperbolic system of PDE: The resolution of this set of PDE constitutes the Darboux method.
In this Section, we will not follow the method given in [22]. We will base the resolution of (3.10)-(3.12) on an intermediary PDE from which the solutions of (3.10)-(3.12) will be deduced. Afterwards, we will construct for any fixed ε map Υ. Then, we will show that Υ is well a change of coordinates and study its regularity with respect to ε. Finally, we will prove Theorem 1.2 and in view of the last Section we will give estimates related to the expression of the characteristics expressed in the Darboux Coordinate System. 11

An intermediary equation
The intermediary equation that we consider in this Section is the following: and where Λ is the vector field defined by: We denote by G λ its flow and by Λ n¨i ts iterated application acting on regular functions f as In a first place, we give the regularity property of G λ .
Theorem 3.2. The unique solution ϕ to (3.13) is given by Proof. The proof of Theorem 3.2 is performed with the usual characteristics' method. Let F pv, s, x, θq be the characteristic associated with (3.13), i.e. the solution of By definition the flow G λ of Λ satisfies: Then, we deduce that F pv, s, x, θq " G εpv´sq px, θq. Eventually Duhamel's formula yields: This ends the proof of Theorem.
We will end this Section by giving a Taylor expansion, with respect to ε, of the solution ϕ to (3.13). Such kind of Taylor expansions are usually referred in the literature (see Olver [27]) as Lie expansions.
where pΛq l is defined by (3.16) and (3.17), and the partial Lie Sum of order n: It is known that, formally, the flow G λ associated with Λ may be expressed in terms of the Lie Series of Λ: More rigorously, as the flow is complete, using its partial Lie Sum we have for any function f : Taking now 1 B as function f and´εv as parameter λ in (3.25), we obtain B˙˝G u px, θq du.

(3.26)
Hence we have proven the following lemma.
Lemma 3.4. Function ϕ, solution to PDE (3.13), admits for any n P N, for any ε P R and for any px, θ, vq P R 4 the following expansion in power of ε ϕ px, θ, vq " B˘˝G´ε u px, θq du is in C 8 # pR 5 q; and for any v P R and any n P N,

The other equations
In the following Theorem, we will deduce from Theorem 3.2 the solutions Υ 1 , Υ 2 , and Υ 4 of the PDEs that are in the left in equalities (3.10)-(3.12).
Theorem 3.5. The unique solutions Υ 1 , Υ 2 , and Υ 4 of are given by where ψ is defined by: with ϕ given by (3.18).
Proof. We will only prove Formula (3.31). The others ((3.32) and (3.33)) are easily obtained with similar arguments. Firstly, we notice that (3.28) can be rewritten as Secondly, integrating (3.13) between 0 and v we obtain Hence by linearity, Υ 1 given by (3.31) is solution of (3.35). The unicity is obvious.
To end the resolution of (3.10)-(3.12) we only have to check that Υ 1 and Υ 2 given by (3.31) and (3.32) are also solutions to the additional equations that are in the right in (3.10)-(3.12).
As the last step of this proof, because of the Jacobi identity we have which reads, because the gradient of a constant is zero, because, according to (3.33), tΥ 4 , Υ 3 u " 1 ε and, as we just saw, because Υ 1 given by (3.31) satisfies tΥ 3 , Υ 1 u " 0, By continuity of the left hand side of (3.45) on R 4 , we deduce that equality (3.45) is valid on R 4 . As tΥ 1 , Υ 4 u may be smoothly extended by 0 in v " 0, and as the unique solution of (3.45) satisfying the boundary condition tΥ 1 , Υ 4 u px, θ, 0q " 0 is zero, we deduce that Υ 1 given by (3.31) satisfies tΥ 1 , Υ 4 u " 0 for all px, θ, vq. Hence (3.37) follows. The proof that Υ 2 , defined by (3.32) and solution of (3.29), is solutions of (3.38) is very similar. This ends the proof of Theorem 3.6.

The Darboux coordinate system
In subsection 3.3 we solved equations (3.10)-(3.12), with initial conditions (3.7), on R 4 . Now, we need to check that the restriction of Υ to R 2ˆRˆp 0,`8q, also denoted by Υ, is a diffeomorphism (onto R 2ˆRˆp 0,`8q) and hence that py, θ, kq makes a true coordinate system on R 2ˆRˆp 0,`8q.
In order to show that py, θ, kq makes also a coordinate system we will proceed as follows: we will express Υ 4 in the py, θ, vq-coordinate system and using this expression, we will express v in terms of y and θ and the yielding expression of Υ 4 in the py, θ, vq-coordinate system.

Proof.
As for all py, θq. Moreover, according to formula (3.50) we have for any v ą 0 the following estimates: and consequently for any py, θq P R 3 rη py, θqs pp0,`8qq " p0,`8q . Particularly, for any v P p0,`8q there exists k P p0,`8q such that v " rη py, θqs´1 pkq . (3.58) The regularity ofη with respect to k is easily obtained from the fact that rη py, θqs is a C 8diffeomorphism. The C 8 -nature ofη with respect to y and θ is obtained by computing the successive derivatives of rη py, θqs˝rη py, θqs´1 " id and using the regularity of η that comes from the regularity ofΥ 4 , itself coming from the regularity of B and flow G λ . Moreover, the periodicity ofη with respect to θ comes from the fact that θ Þ Ñ`G 1 λ px, θq, G 2 λ px, θq˘is in C 8 per pRq (see Notation 1) for any x P R 2 as set out in Lemma 3.1.
Hence we have proven the following theorem.
Lemma 3.11. Function py, θ, k, εq Þ Ñ rβ py, θ, kqspεq , The proof of the periodicity with respect to the third variable is similar to the one of Lemma 3.8.
We will now use Formula (3.64), Lemmas 3.10 and 3.11 to deduce an expression of the expansion with respect to ε of the v-component of κ " Υ´1.

(3.71)
Moreover, thanks to formula (3.72), the P n can easily be computed by induction.
Proof. Proof of Lemma 3.12 is easily done by induction. Notice that the inductive formula for P n is given by: P n`1 pX 1 , . . . , X n`1 q "´p2n´1q X 2 P n pX 1 , . . . , X n qǹ ÿ k"1 Hence finding an expansion of κ v remains to find the successive derivatives of " γ py, θ, kq ‰ evaluated at ε " 0. The following lemma and its proof constitute a constructive way to compute them. Proof. On the one hand, for any py, θ, kq and for any n P N, rγ py, θ, kqs admits a Taylor-MacLaurin expansion of order n.
The two previous Lemmas and Formula (3.64) lead to the following Theorem.
Applying Theorem 3.14, up to order 2, we obtain whereâ "â pθq is defined byâ pθq "ˆc os pθq sin pθq˙ ( 3.80) and where H B is the Hessian Matrix of B.
Remark 3.16. Formula (3.79) can already be found in Littlejohn [22] but without estimation of the rest. In the present paper formula (3.79) gives an expansion in power of ε of a well defined diffeomorphism even though it is obtained in Littlejohn [22] by truncating a formal Hilbert expansion.
As a conclusion, tΥ 1 , Υ 2 u x,θ,v " u, and u is given by (3.82). Hence the Theorem is proven.

89)
where function ι N`1 is in C 8 # pR`ˆR 2ˆRˆp 0,`8qq. Moreover, for any n P t1, . . . , N u there exists a function b n P O 8 T,b such that H n py, θ, kq " ? k n`2 b n py, θq . For instance up to order 2 we obtain: Corollary 3.20. The Hamiltonian function in the Darboux Coordinate System admits, up to order 2, the following expansion in power of ε: whereâ is defined by (3.80), function ι 3 is in C 8 #,3 pR 2ˆRˆp 0,`8qˆR`q, and where H B stands for the Hessian matrix associated with B.
Concerning Lemma 3.24, for any px, θq P R 2ˆR and for any v P ra, bs , function ψ satisfies |ψ px, θ, vq| ď b. Applying formula (3.39) yields: B˙p G´ε u px, θqq du. For any px, θq P R 2ˆR , for any v P ra, bs and for any ε P R we have:ˇˇΥ b 1 px, θ, vqˇˇď and consequently for any px, θq P R 2ˆR , for any v P ra, bs , for any ε P R ‹ and for any t P ŘˇˇΥ b 1`X ε Pol pt; x, θ, vq , Θ ε pt, x, θ, vq , v˘ˇˇď On another hand, evaluating Υ s 1 in´X ε Pol pt; x, θ, vq , Θ ε pt; x, θ, vq , v¯and differentiating with respect to t yields: and consequentlyˇˇˇˇB

The Partial Lie Transform Method
The last step on the way to build the Guiding-Center Coordinates of order N is to build a coordinate system pz, γ, jq close to the Historical Guiding-Center coordinate system in which the Poisson Matrix and the Hamiltonian function are given by (1.13) and (1.14). To this aim we will construct a new algorithm, the so-called Partial Lie Transform Method.
Remark 4.1. In [22], to build the Guiding-Center coordinate system, Littlejohn construct a normal form theory based on formal Lie series using Hamiltonian vector fields. The drawback of using such a formal Lie Series method is that its convergence is neither ensured nor controlled.

The Partial Lie Change of Coordinates of order N
We start this Section by defining the partial Lie sums. Let N P N˚. For i P v1, N w, we define the positive integer α i,N by where E stands for the integer part. be the differential operator acting on functionsf "f py, θ, kq of C 8 # pR 2ˆRˆp 0,`8qq in the following way: where X έ εḡ is the Hamiltonian vector field associated with´εḡ. From operator ϑ α i,N ,i ε,´ḡ we define, with the same notation, function ϑ where y 1 , y 2 , θ, k stand for y 1 : py, θ, kq Þ Ñ y 1 , . . ., k : py, θ, kq Þ Ñ k.  Then there exists η ą 0 such that for any ε P r´η, ηs, χ N ε , defined by is well defined on R 3ˆp c, dq and is a diffeomorphism. Moreover, for any intervals pc ‹ , d ‹ q and pc ‚ , d ‚ q such that c ‹ ą 0 and rc ‚ , d ‚ s pc, dq rc, ds pc ‹ , d ‹ q (4.5) there exists a real number η ‚,‹ ą 0 such that for any ε P r´η ‚,‹ , η ‚,‹ s:  Remark 4.6. An immediate Corollary to Theorem 4.4 is that for ε small enough λ N ε is well defined on R 3ˆp c ‚ , d ‚ q.

Main properties of the partial Lie change of coordinates of order N
The main properties of the partial Lie change of coordinates of order N are summarized in the following Theorem.

The Partial Lie Change of Coordinates Algorithm
In this Section, we will deduce from Formula (4.8) the Partial Lie Change of Coordinates Algorithm. and getḡ i and by solving:

Proof of Theorem 4.4
The first step to prove Theorem 4.4 consists in proving that the partial Lie sums are diffeomorphisms and to localize their ranges.
Theorem 4.11. Let i P v1, N w,ḡ i P Q 8 T,b and c and d be positive real numbers (with c ă d). Then there exists η ą 0 such that for any ε P r´η, ηs, function ϑ defined by (4.3), is a diffeomorphism from R 3ˆp c, dq onto its range. Moreover, for any interval pc ‹ , d ‹ q and pc ‚ , d ‚ q such that c ‹ ą 0 and there exists a real number η ‚,‹ ą 0 such that for any ε P r´η ‚,‹ , η ‚,‹ s: Proof. In a first place, we will show that ϑ α i,N ,i ε,´ḡ i is a diffeomorphism from R 3ˆp c, dq onto its range. To this aim, we will check that there exists a real numberη 1 such that for any ε P r´η 1 ,η 1 s , the map ϑ α i,N ,i ε,´ḡ i satisfies the assumptions of the classical global inversion Theorem.
The second part of the proof concerns inclusions (4.20). Using Formula (4.22) we obtain easily the second inclusion. Hence, we will focus on the first one. Its proof is based on the Brouwer Theorem (see Brouwer [2] or Istratescu [19]).
We fix two positive real numbers R 1 0 and R ‚ 0 such that R ‚ 0 ă R 1 0 . Then we will fix m 0 P R 2 and we will show that there exists a positive real number η, that does not depend on m 0 , such that for any ε P r´η, ηs 28) or according to Notation 12, Consequently, since η does not depend on m 0 we will obtain (4.20).
0 , c p2q , c p3q , d p2q and d p3q be real numbers satisfying l be an integer, and let K l 2 and K l 3 be the compact and convex subsets of R 2ˆRˆp 0,`8q defined by Ñ 0 when ε Ñ 0, we can define η ą 0 (that depends neither on l nor m 0 ) such that for any ε P r´η, ηs, for any l P Z, and for any py 1 , θ 1 , k 1 q P K l 2 ,ˇy 1´m 0ˇ`| ε| (4.32)

Proof of Theorem 4.7
We begin by giving and proving preliminary results that are needed for the proof of Theorem 4.7. Its proof is then led in the last part of this subsection.
Property 4.14. Let i P v1, N w and f ,ḡ i and h be three functions in C 8 #`R 3ˆp 0,`8q˘. Then, the following equalities hold true on R 3ˆp 0,`8q: where ρ N,i F P and ρ N,i Proof. The proofs of Formulas (4.38) and (4.39) are very similar. Consequently, we will only give the proof of Formula (4.39).
In a first place, starting from is the following equality which is a direct consequence of the Jacobi identity, it is obvious to show by induction that (4.41) Secondly, we will define on R 3ˆp 0,`8q the function tf, huT ε " tf, huT ε py, θ, kq by tf, huT ε py, θ, kq "`T ε py, θ, kq ∇hpy, θ, kq˘¨p∇f py, θ, kqq , (4.42) whereT ε " εP ε , (4.43) and notice that ε Þ Ñ tf, huT ε py, θ, kq is in C 8 pRq for any py, θ, kq P R 3ˆp 0,`8q. Hence, expanding ϑ , and making the difference between these two expansions yields (4.39) with ρ N,i P C pε,¨q " As iα i,N ě N`1, all k ě α i,N`1 satisfy ik ě N`2. Consequently, ε Þ Ñ ρ N,i P C pε; y, θ, kq is in C 8 pRq for any py, θ, kq P R 3ˆp 0,`8q. In addition, py, θ, kq Þ Ñ ρ N,i,j P C pε, y, θ, kq is clearly in C 8 #`R 3ˆp 0,`8q˘for any ε P R. (4.45) Theorem 4.16. With the same notations and under the same assumptions as in Theorem 4.7, let i P v1, N w and h ε be in Q 8 T,b X A`R 2ˆRˆp 0,`8q˘for every ε in some interval I containing 0 and such that ε Þ Ñ h ε prq is in C 8 pIq for anyr P R 3ˆp 0,`8q. Then, there exists a real number η K ą 0 such that for any ε P r´η K , η K s X I and for any py, θ, kq P KˆRˆ"c ♦ , d ♦ ‰ , we have Proof. Since h ε P Q 8 T,b , and by linearity, the proof of the theorem reduces to prove formula (4.46) with function h ε of the form h ε py, θ, kq " cos l pθq sin m pθq d ε pyq ?
Letr 0 " py 0 , θ 0 , k 0 q P KˆRˆ"c ♦ , d ♦ ‰ . As d ε P A`R 2˘, and as´k Þ Ñ ? k n¯P App0,`8qq, there exists a real number Rr 0 ą 0 and a formal power series Tr 0 of three variables whose set of convergence contains the closure of b 3`0 , Rr 0˘, which are such that b 3 ppy 0 , k 0 q , Rr 0 q Ă R 2ˆp c ‚ , d ‚ q and such that for any py, kq P b 3 ppy 0 , k 0 q , Rr 0 q, d ε pyq ? k n " Tr 0 ppy, θq´py 0 , θ 0 qq " ÿ lPN 3 a l,r 0 ε ppy, kq´py 0 , θ 0 qq l . (4.48) In addition, since`θ Þ Ñ cos l pθq sin m pθq˘is a power series of convergence radius`8 with respect to θ, there exists a formal power series Sr 0 such that b # p0, Rr 0 q Ă Σ Sr 0 and such that @r " py, θ, kq P b # pr 0 , Rr 0 q , h ε prq " Sr 0 pr´r 0 q " Let R 1r 0 P p0, Rr 0 q. Then, using similar arguments as in the proof of Theorem 4.4 we easily obtain that there exists a real number η Rr 0 ,R 1r 0 ą 0 such that for any ε P "´η On another hand, let Θ ε " Θ ε prq " pΘ ε,m prqq mPN 4 s.t. |m|ďi be the smooth function that are such that, for all smooth functions f ε , Bf ε Br m prq .  where r lr 0 stand for the functionr Þ Ñ pr 1´pr0 q 1 q l 1 pr 2´pr0 q 2 q l 2 pr 3´pr0 q 3 q l 3 pr 4´pr0 q 4 q l 4 .
Since b # p0, Rr 0 q Ă Σ Sr 0 , we can permute summation and derivations and we obtain:  Finally, as

Proof of Theorem 4.8
Having expansion (3.89) in mind, the proof of Theorem 4.8 consists essentially in ordering the terms in Formula (4.8) with respect to their power of ε. More precisely, we will focus on expanding ϑ α 1,N ,1 ε,ḡ 1¨ϑ Applying Lemmas 3.23, 3.24 and 3.25 and because the Lie change of coordinates is close to the identity (see formula (4.22)), it is clear that there exists a compact set K L , positive real numbers c L and d L , and a positive real number η K L such that for any ε P r0, η K L s, for any t P r0, T s, and for any px 0 , v 0 q P K CˆC pc C , d C q, the characteristic pZ, Γ, J q associated with the Hamiltonian system (1.1)-(1.2) and expressed in the pz, γ, jq coordinate system stays in K LˆRˆp c L , d L q. Consequently, we can apply Theorem 4.7.
To end this proof we will prove estimate ( Now, we will check that pZ, J q is in C N´1 pr0, η K L sq. In order to check this, we define for any ε P p0, η K L s , for any t P r0, T s, and for any pz, γ, jq P U C , p r Z, r Γ, r J q bÿ Since ε Þ Ñ εP ε is in C 8 pr0, η K L sq, the solution of (4.81) depends smoothly on the parameter ε. In particular function p r Z, r Γ, r J q, defined by (4.80), is smoothly extensible at ε " 0. On another hand, for any ε P p0, η K L s , and for any t P r0, T s , pZ, J q is solution to are 2π-periodic and smooth, and consequently C 8 b pRq with respect to the third variable γ. Hence, computing the successive derivatives of (4.82) with respect to ε, we obtain that ε Þ Ñ pZptq, J ptqq is C N´1 in the neighborhood of ε " 0.  Z T , J T˘i s smooth with respect to ε, for any t P r0, T s. Now, we will show that L ε defined for ε P p0, η K L s by is extensible to r0, η K L s and that the yielding extension is continuous with respect to ε. By definition for any ε P p0, η K L s, for any t P r0, T s, ε Þ Ñ L ε is C N´1 pp0, η K L sq. So, we just have to show that ε Þ Ñ L ε is extensible as a continuous function on r0, η K L s, i.e. that ε " 0 is not a singularity. In a first place, for any ε P p0, η K L s, we will explicit the dynamical system L ε satisfies. InjectingˆZ in (4.82) gives (4.89) Making a Taylor expansion in where β 1 is smooth and periodic of period 2π with respect to γ. Injecting (4.90) in (4.89) and using (4.86) yields where β 2 and β 3 are smooth and 2π-periodic with respect to γ. Besides, the solutions of this dynamical system are continuous with respect to ε. Clearly the initial data for L ε is L ε p0q " 0. Hence, L ε is continuous with respect to ε. Since pZ, J q´`Z T , J T˘" ε N´1 L ε , estimate (1.17) follows. This ends the proof of Theorem 1.5.

A Appendix : Change of coordinates rules for the Poisson Matrix and the Hamiltonian Function
A Poisson Matrix P on an open subset of R 4 is a skew-symmetric matrix satisfying: @i, j, k P t1, . . . , 4u , ttr i , r j u , r k u`ttr k , r i u , r j u`ttr j , r k u , r i u " 0, where r i is the i-th coordinate function r Þ Ñ r i and the Poisson Bracket tf, gu between smooth functions f and g is defined by (2.8).
In the case of a symplectic manifold, the Poisson Matrix in a given coordinate system is defined as follow: it is the inverse of the transpose of the matrix of the expression of the Symplectic Two-Form in this coordinate system. Notice that the Jacoby identities (A.1) are direct consequences of the closure of the Symplectic Two-Form.
We now turn to the change-of-coordinates rule for the Poisson Matrix. Firstly, if in a given coordinate chart m, the matrix associated with the Symplectic Two-Form reads K, then, according to the previous definition, the Poisson Matrix is given by P pmq " pK pmqq´T . (A.2) If we make the change of coordinates σ : m Þ Ñ r, then the usual change-of-coordinates rule for the expression of the Symplectic Two-Form leads to the following change of coordinates rule for the Poisson Matrix where i X G dΩ is the interior product of differential two-form dΩ by vector field X G . The expression of the Hamiltonian vector field associated with the Hamiltonian function G, in the coordinate system m, is the vector field which reads: where G is the representative of G in this coordinate system. In fact, we can consider or equivalently as said in the introduction, the dynamical system whose expression in every coordinate system r is given by where G 1 is the representative of G in this coordinate system, and P 1 the expression of the Poisson Matrix in this coordinate system. In particular, if we check that on a global coordinate chart, a dynamical system is Hamiltonian, then the dynamical system is Hamiltonian on M and its expression in every coordinate chart r is given by (A.8).