A Modified Parameterization Method for Invariant Lagrangian Tori for Partially Integrable Hamiltonian Systems

In this paper we present an a-posteriori KAM theorem for the existence of an $(n-d)$-parameters family of $d$-dimensional isotropic invariant tori with Diophantine frequency vector $\omega\in \mathbb R^d$, of type $(\gamma,\tau)$, for $n$ degrees of freedom Hamiltonian systems with $(n-d)$ independent first integrals in involution. If the first integrals induce a Hamiltonian action of the $(n-d)$-dimensional torus, then we can produce $n$-dimensional Lagrangian tori with frequency vector of the form $(\omega,\omega_p)$, with $\omega_p\in\mathbb R^{n-d}$. In the light of the parameterization method, we design a (modified) quasi-Newton method for the invariance equation of the parameterization of the torus, whose proof of convergence from an initial approximation, and under appropriate non-degeneracy conditions, is the object of this paper. We present the results in the analytic category, so the initial torus is real-analytic in a certain complex strip of size $\rho$, and the corresponding error in the functional equation is $\varepsilon$. We heavily use geometric properties and the so called automatic reducibility to deal directly with the functional equation and get convergence if $\gamma^{-2} \rho^{-2\tau-1}\varepsilon$ is small enough, in contrast with most of KAM results based on the parameterization method, that get convergence if $\gamma^{-4} \rho^{-4\tau}\varepsilon$ is small enough. The approach is suitable to perform computer assisted proofs.


Introduction
Persistence under perturbations of regular (quasi-periodic) motion is one of the most important problems in Mechanics and Mathematical Physics, and has deep implications in Celestial and Statistical Mechanics.The seminal works of Kolmogorov [19], Arnold [1] and Moser [22] put the name to the KAM theory, that has become a full body of knowledge that connects fundamental mathematical ideas in many different contexts around the so-called small divisors.See e.g. the popular book [9], and the surveys [2,6].
Although KAM theory held for general dynamical systems under very mild technical assumptions, its application to concrete systems became a challenging problem.With the advent of computers and new methodologies, the distance between theory and practice was shortened (see e.g.[4,8,5]).One direction that have experienced a lot of progress is the a-posteriori approach based on the parameterization method [6,7,16,28,12], that in this context was originally known as KAM theory without angle-action coordinates.This approach lead to the design of a general methodology to perform computer assisted proofs of existence of Lagrangian invariant tori [10], enlarging the threshold of validity to practically the one predicted by numerical observations [15,20], in academic examples such as the Chirikov stardard map.But new impulses have to be made in order to make KAM theory fully applicable to realistic physical systems, which often have extra first integrals, or degeneracies.
The goal of this paper is to present a KAM theorem in a-posteriori format for the existence of invariant Lagrangian cylinders for Hamiltonian systems with first integrals in involution, see Theorem 2.18.If the first integrals induce a Hamiltonian action of a torus, then the theorem produces invariant Lagrangian tori.The presence of first integrals in involution is usually treated with symplectic reduction techniques [3,21], and then applying KAM theorems to the reduced systems.For the sake of versatility, we do not pursue such changes of variables, and we get n-dimensional Lagrangian invariant cylinders from a single d-dimensional isotropic torus through a Reduction lemma, see Lemma 2.5, that avoids the use of symplectic reduction techniques.Of course, our results work for systems without additional first integrals in involution, that could be obtained after reduction techniques.
Although the setting of the paper is close to that in [17], we incorporate some constructs in [30] to improve crucial estimates.We present a (modified) quasi-Newton method for these systems.The method solves exactly the same equations as in the quasi-Newton method in [17], but differ in the last step with [17] in the way the updates of the parameterization of the torus are made.In particular, normal corrections to the torus are made to improve the invariance of the torus, while tangent corrections to the torus are made to conjugate the internal dynamics of the torus to a linear flow.As a result, while in [17] we got convergence of the quasi-Newton method if γ −4 ρ −4τ ε is small enough, as in most of KAM papers based on the parameterization method, in this paper we get convergence if γ −2 ρ −2τ −1 ε is small enough, as in [30], [31] (see also the prequel [29]), and in the KAM Theorem [1,24,25].But, while the inspiring paper [30] substitutes the invariance equation of the parameterization method by three different conditions which are altogether equivalent to invariance, here we suitably project the error of invariance in tangent and normal components, as in [17].Also, our proof of the result heavily relies on geometrical and reducibility properties, complementing the approach in [30], that does not use (at least explicitly) these properties.
In order to emphasize the geometric properties, and for the sake of versatility, we have considered symplectic structures other than the canonical.This was one main point in the seminal paper [7].Here, however, we have only considered symplectic structures that have a compatible almost-complex structure, i.e. that is inducing a Riemannian metrics, as in [14].It is known that any symplectic manifold admits a compatible almost-complex structure (see e.g.[3]).
This paper complements [17] also in the sense that, from the algorithm derived in that paper (whose convergence is proved), one can produce approximations of the solutions of the invariance equations, and then apply the a posteriori KAM theorem derived in this paper, in combination with techniques introduced in [10].There are situations in which the a posteriori KAM theorem in [17] could fail to be applied in practical situations, in which computer resources time and memory are finite.For instance, when the tori are high dimensional and/or about to break, thus needing high order Fourier approximations and/or being the size of the analyticity strip ρ very small.The improved estimates here could mitigate such problems, thus pushing further the domain of existence of KAM tori.
As usual in KAM theory, and in particular in the parameterization method, the proof of existence of invariant tori is pursued by means of a quasi-Newton method in a scale of Banach spaces, here analytic functions.At each step the analyticity strip in which the objects are defined is reduced, and one has to control the whole sequence to get convergence to an object defined in a final analyticity strip.The way the bites are produced on the analyticity strip seems to influence the results in practical situations.We have included a digression on the fact that the best choice seems to be close to consider geometric series of bites with ratio 1  2 .The paper is organized as follows.Section 2 introduces the background and the geometric and analytical constructions, and the main result, Theorem 2. 18, is presented at the end.The proof of Theorem 2.18 is detailed in Section 3.An auxiliary lemma to control inverses of matrices is included in Appendix A. In order to collect the long list of expressions leading to the explicit estimates and conditions of the theorems, we include separate tables in the Appendix B. We pay special attention in providing explicit and rather optimal bounds, with an eye in the application of the theorems and in computer assisted proofs.

The setting and the KAM theorem
In this section we set the geometrical and analytical background of this paper, and present the main result.In Subsection 2.1 we establish the basic notation.In Subsection 2.2 we introduce the geometrical setting of this paper, i.e. symplectic structures on open sets of R 2n that admit compatible almost-complex structures.In Subsection 2.3 we review standard definitions for Hamiltonian systems and first integrals, with an eye in Hamiltonian actions of tori.In Subsection 2.4 we set the main equations of this paper, to find invariant tori carrying quasi-periodic motion, and some implications of the presence of first integrals in involution, that lead to a reduction of the dimensionality of the problem of finding Lagrangian invariant cylinders and tori, without the need of reducing the Hamiltonian itself, see Lemma 2.5.In Subsection 2.5 we review some geometrical constructions and reducibility properties of invariant tori, and present some implications of the presence of compatible triples, see Proposition 2.9.In Subsection 2.6 we review the quasi-Newton method introduced in [17], and present a new modified quasi-Newton method to solve the invariance equations.In Subsection 2.7 we introduce the spaces in which the invariance equations are considered, i.e. spaces of analytic functions, and review some key results regarding small divisors equations, see Lemmas 2.14 and 2.16.Finally, in Subsection 2.8 we present the main result of this paper, Theorem 2.18, an a posteriori theorem on the existence of isotropic invariant tori for Hamiltonian systems with first integrals in involution.The proof of this theorems constitutes the bulk of this paper and is found in Section 3.
2.1.Basic notation.We denote by R m and C m the vector spaces of m-dimensional vectors with components in R and C, respectively, endowed with the norm Given a, b ∈ C, we also often use the notation |a, b| = max{|a|, |b|}.We consider the real and imaginary projections Re, Im : We denote R n1×n2 and C n1×n2 the spaces of n 1 × n 2 matrices with components in R and C, respectively, identifying R m ≃ R m×1 and C m ≃ C m×1 .We denote I n and O n the n × n identity and zero matrices, respectively.The n 1 × n 2 zero matrix is represented by O n1×n2 .Finally, we use the notation 0 n to represent the column vector O n×1 , although we mostly write 0 when the dimension is known from the context.Matrix norms in both R n1×n2 and C n1×n2 are the ones induced from the corresponding vector norms.That is to say, for an In particular, if v ∈ C n2 , |M v| ≤ |M ||v|.Moreover, M ⊤ denotes the transpose of the matrix M , so that Given an analytic function f : U ⊂ C m → C, defined on an open set U, the action of the r-order derivative of f at a point x ∈ U on a collection of vectors v 1 , . . ., v r ∈ C m , with where the indices ℓ 1 , . . ., ℓ r run from 1 to m.This construction is extended to vector and matrix-valued maps as follows: given a matrix-valued map M : U ⊂ C m → C n1×n2 (whose components M i,j are analytic functions), a point x ∈ U, and a collection of (column) vectors v 1 , . . ., v r ∈ C m , we obtain an Notice that, if we split M in its columns M •,j for j = 1, . . ., n 2 , so that (M •,j For r = 1, we will often write DM (x , we can think of Df as a matrix function Df : U → C n×m , hence, D 1 f (x)[v] = Df (x)v for v ∈ C m .Therefore, we can apply the transpose to obtain a matrix function (Df ) ⊤ , which acts on n-dimensional vectors, while Df ⊤ = D(f ⊤ ) acts on m-dimensional vectors.Hence, according to the above notation, the operators D and (•) ⊤ do not commute.Therefore, in order to avoid confusion, we must pay attention to the use of parenthesis.
A function u : R d → R is 1-periodic if u(θ + e) = u(θ) for all θ ∈ R d and e ∈ Z d .Abusing notation, we write u : and e ∈ Z d .We also abuse notation and write u : |Im θ| < ρ} is the complex strip of T d of width ρ > 0. We write the Fourier expansion of a periodic function as and introduce the notation u := û0 for the average.Given ω ∈ R d , we define the operator L ω acting on u as the Lie derivative of u in the direction of the constant vector field θ = −ω on the torus: (2.1) The notation in this paragraph is extended to n 1 × n 2 matrix-valued periodic maps M : 2.2.Symplectic setting.In this paper we consider the phase space is an open set U of R 2n , whose points are denoted by z = (z 1 , . . ., z 2n ), endowed with an exact symplectic form ω = dα, where the 1-form α is called action form.The setting and the results of this paper can be easily adapted to other settings such as U ⊂ T k × R 2n−k with k ≤ n.See e.g.[16] for a discussion.In order to simplify some of the geometrical constructs of this paper, we also assume that U is endowed with a Riemannian metric g and an anti-involutive linear isomorphism J : TU → TU , i.e.J 2 = −I, such that ∀z ∈ U, ∀u, v ∈ T z U, ω z (J z u, v) = g z (u, v).It is said that (ω, g, J ) is a compatible triple and that J endows U with an almost-complex structure.The anti-involution preserves both 2-forms ω and g.
We rather use the matrix representations of the previous objects, given by the matrixvalued maps a : U −→ R 2n , representing the 1-form α, and Ω, G, J : U −→ R 2n×2n , representing the 2-forms ω and g, and the anti-involution J , respectively.The fact that ω is closed reads ∂Ω r,s for any triplet (r, s, t), and the fact that ω is exact, with ω = dα reads Moreover, Ω ⊤ = −Ω, and Ω is pointwise invertible.Moreover, the metric condition of g reads G ⊤ = G and it is positive definite, and the compatibility conditions read Notice also that Ω = GJ.These properties also imply the relations Remark 2.1.The prototype example of compatible triple is (ω 0 , g 0 , J 0 ) in R 2n , where ω 0 is the standard symplectic structure, ω 0 = n i=1 dz n+i ∧ dz i , g 0 is the Euclidian metric, and J 0 is the linear complex structure in R 2n (as a real vector space), coming from the complex structure in C n .The matrix representations of these objects are Moreover, an action form for ω 0 is α 0 = 1 2 n i=1 (z n+i dz i − z i dz n+i ), which is represented as Remark 2.2.Even though usually KAM theory is presented in the stardard case, a notable counterexample is the seminal paper [7] that was followed by other papers such as [13,14,10].Even more general constructs were presented in chapter 4 of [16].

2.3.
Hamiltonian systems and first integrals in involution.Given a function h : U → R, the corresponding Hamiltonian vector field X h : U → R 2n is the one such that i X h ω = −dh.
In coordinates, the Hamiltonian vector field X h satisfies Using Cartan's magic formula, the Lie derivative of ω in the direction of X h vanishes, The Poisson bracket of two functions f , g is given by {f, g} = −ω(X f , X g ).It is related to the Lie bracket through the well-known formula [X f , X g ] = −X {f,g} , that in coordinates is written as {f, g} = −(X f ) ⊤ Ω X g = Df X g .In particular, f is a first integral or preserved quantity of X g if and only {f, g} = 0, and it is said that f, g are in involution.As a consequence, DX f X g = DX g X f and the corresponding flows commute.
We assume there is a moment map p : U → R n−d , meaning that its components p 1 , . . ., p n−d , jointly with h, are pairwise in involution functionally independent functions.We encode the corresponding Hamiltonian vector fields as the columns of X p : U → R 2n×(n−d) : The properties mentioned above are summarized as follows: Dp X p = 0, Dp X h = 0 (and, then, DX p [X h ] = DX h X p ), and the matrix X h (z) X p (z) has (maximal) rank n − d + 1, for any z ∈ U .
For j = 1, . . ., n − d, the vector fields X pj generate (local) flows ϕ j : D j ⊂ R × U → U , for which we write ϕ j sj (z) = ϕ j (s j , z) for (s j , z) ∈ D j .The flows commute (and also commute with the flow of X h ).We then define Φ : , where we say that Φ is the (local) moment flow associated to X p , and write Φ s (z) = Φ(s, z).Notice also that D z Φ X p = X p •Φ, and D z Φ X h = X h •Φ.The case d = 1 corresponds to the integrable case, and from now on we will assume d > 1.
The case d = n corresponds to not assuming the existence of first integrals in involution, and all the results of this paper hold for such a case.Remark 2.3.An important special case is when the moment map p induces a Hamiltonian torus action, that is D = R n−d × U and the generated flow Φ : R n−d × U → U is periodic in all components of s.By scaling times we can get all periods equal to one, and then, with a slight abuse of notation, consider Φ : 2.4.Invariant tori.In this paper, we refer to an embedding K : T d → U as a parameterization of the torus K = K(T d ).
Given ω ∈ R d with 1 ≤ d ≤ n , we say that K is invariant for X h with frequency vector ω if (2.3) This means that the d-dimensional torus K = K(T d ) is invariant and the internal dynamics is given by the constant vector field θ = ω.Equation (2.3) is called invariance equation for K and frequency ω.Case d = 1 corresponds to K being a periodic orbit.We then assume d ≥ 2 and ω to be ergodic: for k ∈ Z d \{0}, k • ω = 0. Hence, the flow on the torus is quasi-periodic and non-resonant.
Remark 2.4.Other topologies can be considered for both the ambient manifold U and the torus K. See e.g.[16] for a discussion.
Ergodicity of ω implies additional geometric and dynamical properties of the torus K.In particular, it is isotropic, i.e. the pullback K * ω is zero (see [18,23]), and it is contained in an energy level of the Hamiltonian and of the additional first integrals (if any): Another consequence of the presence of extra first integrals in involution is that an invariant torus K, with frequency vector ω, induces a family of invariant tori K s = Φ s (K), with the same frequency vector (with s defined, a priori, in an open neighborhood We can think the family foliating an n-dimensional invariant object K parameterized by K : Notice that we can rephrase the previous argumentation by writing This argument of getting n-dimensional invariant objects (including n-dimensional invariant tori) from d-dimensional invariant tori is extended in the following lemma.
Lemma 2.5 (Reduction lemma).Let h : U → R be a Hamiltonian for which there exists a moment map p : R be a function defined on the set of possible momenta, and let ĥ : U → R be the discounted Hamiltonian, defined as ĥ = h − f •p.Let K : T d → U be a parameterization of a torus K = K(T d ), invariant for X ĥ with ergodic frequency ω ∈ R d , i.e. (2.4) and Proof.The hypotheses (2.4) of invariance of K for the vector field X ĥ reads Then, since Let us assume now that K(θ 0 ) = K(θ 0 , S) = Φ S (K(θ 0 )).Denote φt the (local) flow of X ĥ.Then, for all t ∈ R, and the result follows from ergodicity of ω.
Remark 2.6.We can think of K as the generator of the n-dimensional object K.The generator is defined up to a change of phase, say α ∈ R d , and flying times, say β ∈ B, since the parameterization K α,β defined as is also a generator of K.This indeterminacy of the generator can be fixed by imposing extra conditions apart from the invariance equation.Some strategies are described in [17].
Remark 2.7.In case of existence of θ 0 ∈ T d such that K(θ 0 , S) = K(θ 0 ) for a certain S ∈ R n−d , Lemma 2.5 establishes the existence of an n-dimensional invariant torus K, with frequency vector ω = (ω, ω p0 /S) (we understand here that the division of the two vectors is made componentwise).In other words, by scaling periods we get a parameterization K : (again considering the product Sϑ componenwise), that satisfies Remark 2.8.In case the moment map p induces a Hamiltonian torus action, see Remark 2.3, we can consider the moment flow as a map Φ : T n−d ×U → U (this is in fact the Hamiltonian action of T n−d on U ).In particular, for a fixed complementary frequency ω p ∈ R n−d , to which we will refer to as a moment frequency, we consider the discounted Hamiltonian ĥωp = h − p ⊤ ω p .Then, a parameterization K : T d → U invariant for X ĥωp with frequency ω induces a parameterization K : T n → U invariant for X h with frequency ω = (ω, ω p ).We can also think of the moment frequency ω p as a parameter, and the discounted Hamiltonian ĥωp as a family of Hamiltonians.From the results of this paper, one can obtain n − d parametric families of n-dimensional tori with fixed internal frequency ω, parameterized by ω p .
2.5.Linearized dynamics and reducibility.In this section we describe the geometric construction of a suitable symplectic frame attached to a torus K with respect to a Hamiltonian system X h , possibly with first integrals p, and an ergodic frequency ω ∈ R d .
In the following, for a matrix-valued map V : If we think of V as a parameterization of a frame of an m-dimensional vector bundle V, then X V corresponds to its infinitesimal displacement by the flow of X h around K. We say that V is invariant under (the linearized equations of) X h if X V = O 2n×m .We also define the pullback of Ω and G of V on K to be the matrix-valued maps Ω V , G V : We consider the matrix-valued map L : and we assume that rank L(θ) = n for every θ ∈ T d .Notice that, while DK parameterizes the tangent bundle of K, T K , L parameterizes a subbundle L of rank n of the bundle T K U .We refer to L as the tangent frame (attached to K).
One can use the geometric structure of the problem to complement the above frame.From the several choices one could do (see e.g.[16] for a discussion), we consider here the one that is specially tailored for a compatible triple.In particular, we define N : Notice that N parameterizes another subbundle N of rank n of T K U .We refer to N as the normal frame (attached to K).
Finally, we define the matrix-valued map P : T d → R 2n×2n as the juxtaposition of matrix-valued maps L, N : We refer to P as an (adapted) frame (attached to K).
We define then the torsion of the parameterization K (with respect to the Hamiltonian h) to be the matrix-valued map T : T d → R n×n defined as (2.9) where T h : U → R 2n×2n is the torsion of the Hamiltonian h, defined as where we use that and G = −ΩJ = J ⊤ Ω.Notice that T h is symmetric: As a result, the torsion T of the parameterization K is symmetric.The use of the frame P has several advantages.Among them, it produces a natural and geometrically meaningful non-degeneracy condition (twist condition, that is the invertibility of the average of the torsion T ) in the KAM theorem, and, most importantly, when the torus is invariant, it reduces the linearized dynamics to a block-triangular form.
Proposition 2.9.If K is invariant for X h with ergodic frequency ω, then: (1) P is symplectic:

and, in particular, L and N parameterize complementary Lagrangian bundles (T
where T is defined in (2.9), so Proof.The facts that L is invariant, i.e.X L = O 2n×n , and Lagrangian, i.e.Ω L = O n , follow directly from the invariance of K and the fact that the frequency ω is ergodic.The construction leads to the Lagrangianity of N , i.e.Ω N = O n , and the fact that P is symplectic.
The construction also leads to the reducibility property (2.11) with See e.g.[17].Finally, T = N ⊤ T h N follows from the computation: Remark 2.10.The torsion measures the symplectic area determined by the normal bundle and its infinitesimal displacement.Notice that, in the present paper, the torsion involves geometrical and dynamical properties of both the torus and the first integrals.
2.6.A (modified) quasi-Newton method.In this section we outline a (modified) quasi-Newton method to obtain a solution of the invariance equation (2.3) from an initial approximation.Sufficient conditions of the convergence of the method are provided in Theorem 2.18, whose proof is detailed in Section 3. We focus here on the geometry of the method.Assume then we are given a parameterization K : T d → U .The error of the invariance is E : We may obtain a new parameterization K = K + ∆K by considering the linearized equation If the error E is sufficiently small and we obtain a good enough approximation of the solution ∆K of (2.12), then K provides a new parameterization with a new error Ē which is quadratically small in terms of E. To do so, following the nowadays standard practice [7,16] we may resort to a frame P (here the one defined in 2.8), so by writing We think of the components of ξ as the tangent and normal components of the correction.The new approximation is Following e.g.[14,17], using (2.13), multiplying both sides of (2.12) by Ω -1 0 P ⊤ Ω•K (an approximation of P -1 ), and skipping second order small terms (in the error E and correction ξ) one reaches the block-triangular system ) and (2.9)), where as the tangent and normal components (of the negative) of the error E.
Inspired by [30], we may also consider the new approximation as where the unknowns are and id is the identity map on T d .By Taylor expanding the previous expression, we get where, as usual, hot stands for higher order terms.Hence, the approximations (2.17) and (2.14) differ in quadratically small terms, and the correction terms ξ = (ξ L , ξ N ) may be computed by solving the triangular system (2.15).
It turns out that the triangular system (2.15), requires to solve two cohomological equations consecutively.More specifically, the system is where and the torsion T is given by (2.9).
In the solution of the triangular system it is crucial the fact that the average of the normal component of the error, η N , is zero.Notice that The fact that Ω DK = O d follows directly from the exact symplectic structure, since K * ω = d(K * α), from where η N = 0 follows immediately.See [17].
Quantitative estimates for the solutions of such equations (under Diophantine conditions of ω) are obtained by applying Rüssmann estimates, see Lemma 2.14.Here we just mention that if we denote by R ω v the only zero-average solution u of equation L ω u = v − v , then, since η N = 0 (see the compatibily condition above) and T is invertible (the so-called twist condition), ξ = (ξ L , ξ N ) is given by is the solution of the system (2.18),(2.19)with ξ L = 0 n .While the average of the normal correction is selected to solve the equation for ξ L , and it is ξN we have the freedom of choosing any value for the average of the tangent correction, ξ L = ξL 0 ∈ R n .This is related with the freedom to select a particular generator of the ndimensional torus, which is a d-dimensional torus, and a particular phase, see Remark 2.23.For the sake of simplicity, we select the solution with ξL 0 = ξ L = 0 n .Recapitulating, for future reference we describe one step of the (two) quasi-Newton methods described above as follows: (1) Compute the error of invariance: (2) Compute the tangent and normal frames to the torus: (3) Compute the tangent and normal components of the error of invariance: (4) Compute the torsion: where (5) Compute the tangent and normal components of the correction, provided that T is invertible: ξN where R ω is the Rüssmann operator (see Lemma 2.14).( 6) Compute the new approximation as: • (quasi-Newton method) As it is usual in the a-posteriori approach to KAM theory, the argument consists in refining an initial approximation K by means of the iterative method and proving the convergence to a solution of the invariance equation.The proof of the corresponding theorem to the quasi-Newton method is in [17].The goal of this paper is proving such a result for the modified version.
Remark 2.11.We have seen that the difference of the two approaches (2.14) and (2.17) are quadratically small.As we will see, the way the correction ξ L is included in (2.17) results in a better behavior of the analyticity properties.This is the main idea in [30], in which the invariance equation is replaced by three conditions which are altogether equivalent to invariance.Instead, we will keep the invariance equation formulation.
Remark 2.12.An important feature of the quasi-Newton method is that it can be implemented in a computer.See e.g.[16] for some details of implementations in similar contexts, using FFT.In this respect, it seems that the approach (2.14) is better than (2.17), since the second involves compositions of periodic functions that, in general, are approximated by truncated Fourier series.Composition of periodic functions is much harder computationally than multiplications of periodic functions, that can be done using Fast Fourier Transform.Remark 2.13.Approaches as (2.14) have been implemented as computer assisted proofs [10], for invariant KAM tori for exact symplectic maps.We think that the new approach (2.17) could result in better posed analytical bounds, that could improve the efficiency of the computer assisted proofs.2.7.Analytic setting.The proof of the convergence of the algorithm is presented in the analytic category.Hence, we work with real analytic functions defined in complex neighborhoods of real domains.We consider the sup-norms of (matrix-valued) analytic maps and their derivatives (see the notation in Section 2.1).That is, for f : U ⊂ C m → C, we consider and notice, of course, that the norms M ⊤ U and D r M ⊤ U are obtained simply by interchanging the role of the indices i and j.
The above norms present Banach algebra-like properties.For example, given is also analytic and satisfies There is also a similar bound for the action of the transpose: In addition, given In particular, if f : U ⊂ C m → C n1 , we may take n 2 = 1 and consider the matrix constructions just made.One recover bounds such as Notice we obtain two possible upper bounds for (Df V ) ⊤ U .Spaces of periodic real-analytic functions.The particular case of real-analytic periodic functions deserves some additional definitions and comments.We denote by A(T d ρ ) the Banach space of holomorphic functions u : T d ρ → C, that can be continuously extended to Td ρ , and such that u(T d ) ⊂ R (real-analytic), endowed with the norm We also denote by A C r (T d ρ ) the Banach space of holomorphic functions u : T d ρ → C whose partial derivatives up to order r can be continuously extended to Td ρ , and such that u(T d ) ⊂ R, endowed with the norm u ρ,C r = max k=0,...,r As usual in the analytic setting, we use Cauchy estimates to control the derivatives of a function.Given u ∈ A(T d ρ ), with ρ > 0, then for any 0 < δ < ρ the partial derivative ∂u/∂θ ℓ belongs to A(T d ρ−δ ) and we have the estimates The above definitions and estimates extend naturally to matrix-valued maps, that is, given As it was mentioned in Section 2.1, the operators D and (•) ⊤ do not commute.In particular, given a real analytic vector function w : T d ρ → C n ≃ C n×1 , we have: Small divisors equations and Diophantine vectors.Another ingredient in KAM theory are the estimates of the solutions of the small divisors equations.Given ω ∈ R d , ergodic, and a real-analytic periodic function v ∈ A(T d ρ ), we consider the equation (2.20) where L ω is defined in (2.1).Expanding in Fourier series, the only zero-average solution and all the other solutions of (2.20) are of the form u = û0 + R ω v with û0 ∈ R.
The convergence of the expansion (2.21) is implied by a Diophantine condition on ω.Specifically, for given γ > 0 and τ ≥ d − 1, we denote the set of Diophantine vectors where |k| 1 = d i=1 |k i |, and ω ∈ D d γ,τ .Sharp estimates are provided by the following lemma [10], in which we also include sharp estimates of partial derivatives of the solution, in combination with Cauchy estimates.
where c R (δ) depends on ω and δ, and it is bounded from above by a constant ĉR than depends only on d and τ .More concretely, for any m > 0, Proof.There results follows from the classical results in [26,27], where a uniform bound (independent of δ) is obtained.We refer to [10] for sharp non-uniform computer-assisted estimates (in the discrete case) of the form c R = c R (δ), which represent a substantial advantage in order to apply the result to particular problems.Adapting these estimates to the continuous case is straightforward.Also, we refer to [11] for a numerical quantification of these estimates and for an analysis of the different sources of overestimation.
Remark 2.15.In applications, for a given δ ∈]0, ρ] one selects m big enough so that the integral term in c R (δ, m) or c 1 R (δ, m) is small compared with the preceeding sum of terms up to order m.
Along the proof, we encounter situations in which we have to combine Cauchy and Rüssmann estimates.The following lemma gives sharp bites to perform such combined bounds. (2.22) Proof.The estimates for the derivatives of the solution u follow from applying Cauchy and Rüssmann estimates with bites δ− δ and δ, respectively, and choosing δ to maximize (δ− δ) δτ .This happens for δ = τ 1+τ δ.Remark 2.17.If one applies Rüssmann and Cauchy estimates with bites δ/2 to get the upper bound (2.22), then one obtains Again, the above definitions, constructs and estimates extend naturally to matrix-valued maps.
2.8.The KAM theorem.In this subsection, we present an a-posteriori KAM theorem for d-dimensional quasi-periodic invariant tori in Hamiltonian systems with n degrees-offreedom that have n − d additional first integrals in involution.The hypotheses in Theorem 2.18 are tailored to be verified with a finite amount of computations.and c DΦ such that: the matrix representations Ω, G, J : U → C 2n×2n of ω,g,J satisfy: the Hamiltonian vector field X h : U → C 2n and its torsion T h : U → C 2n×2n , satisfy: the moment vector fields X p : U → C 2n×(n−d) and the moment flow Φ : D → U, satisfy: H 2 There are r > 0, an open subset U 0 ⊂ U, and condition numbers σ DK , σ (DK) ⊤ , σ B , σ N , σ N ⊤ , and σ T -1 such that: , with 0 < ρ < r, is an embedding with K( Td ρ ) ⊂ U 0 , whose averaged torsion T is invertible and, moreover: Under the above hypotheses, for each ρ ∞ ∈]0, ρ[ and δ ∈]0, (ρ − ρ ∞ )/3[, there exists a constant C depending on ρ, ρ ∞ , δ and the constants introduced above such that, if the error of invariance then there exists an invariant torus Furthermore, the objects are close to the original ones: there exist constants the conditions for N ρ and N ⊤ ρ follow inmediatelly.Our point is to provide maximum flexibility of the results to be applied to specific problems.Similar controls could be also do for other objects, such as X p •K, (X p •K) ⊤ , G L or T , leading to similar formulae.Remark 2.20.If d = n then there are no additional first integrals and we recover the classical KAM theorem for Lagrangian tori.The corresponding estimates follow by taking zero the constants c Xp = 0, c X ⊤ p = 0, c DXp = 0, c DX ⊤ p = 0, c DΦ = 1.Remark 2.21.In the canonical case we have Ω = Ω 0 , G = I 2n , and J = Ω 0 and, hence, Remark 2.22.Theorem 2.18 produces a d-dimensional isotropic invariant torus with frequency ω ∈ D d γ,τ , that generates an (n − d)-parameters family of d-dimensional isotropic invariant tori with such a frequency, foliating an n-dimensional invariant cylinder.With the aid of discounted Hamiltonians, one can produce also n-dimensional invariant cylinders, see Lemma 2.5 or, if the moment map p induces a Hamiltonian torus action, one can produce n-dimensional invariant tori, see Remark 2.8.These tori have frequencies (ω, ω p ) ∈ D d γ,τ × R n−d , thus one obtains analytic families of Lagrangian invariant tori.Remark 2.23.The invariant d-dimensional tori are locally unique, meaning that if there is another d-dimensional invariant torus with the same frequency nearby, then both generate the same invariant cylinder.More specifically, the corresponding parameterizations K and K ′ , say, are related by As mentioned in Remark 2.6, both indeterminacies (the phase α and the displacement β) could be fixed by adding n extra scalar equations to the invariance equation.
Remark 2.24.Theorem 2.18 gives the convergence to a parameterization of an invariant torus defined in a complex strip of size ρ ∞ from a parameterization of an approximately invariant torus defined in a complex strip of size ρ, through a sequence of approximations (given by a Newton-like method) whose complex strips sizes are determined by the initial bite 3δ (in the proof, the bites are given by a geometric sequence).In practical situations, these are parameters that can be adjusted appropriately.Heuristically, see Remark 3.7, a good choice is δ = (ρ − ρ ∞ )/6.Also, if one is not interested in controlling the domain of analyticity of the invariant torus, can take ρ ∞ = 0. Since the conditions on the initial parameterization are given by strict inequalities, and the final constants depend continuously on all constants in the hypothesis (including the sizes), then it follows that for a small enough final strip size the conditions hold.

Proof of the KAM theorem
In this section we present a fully detailed proof of Theorem 2.18.Hence, from now on we assume the setting and hyphoteses of Theorem 2.18.The proof consists in demonstrating the convergence of the (modified) quasi-Newton method outlined in Subsection 2.6.In Subsection 3.1 we present some estimates regarding the control of some geometric and dynamical properties for an approximately invariant torus.In Subsection 3.2 we produce quantitative estimates for the objects obtained when performing one iteration of the procedure.Finally, in Subsection 3.3 we discuss the convergence of the (modified) quasi-Newton method.
3.1.Some lemmas to control approximate geometric properties.Here we present some estimates regarding the control of some geometric and dynamical properties for an approximately invariant torus, including approximate symplecticity of the corresponding frame, the control of the total error of invariance by its tangent and normal projections, and the approximate reducibility of the linearized dynamics.We collect all constants appearing in the bounds in Appendix B, Table 1.

Approximate symplecticity of the adapted frame.
We prove here that the adapted frame P : Td ρ → C 2n×2n attached to the torus K parameterized by K : Td ρ → U 0 , defined in (2.8), induces an approximately symplectic vector bundle isomorphism and, in particular, that the bundle L framed by L : Td ρ → C 2n×n given in (2.7) is approximately Lagrangian.See e.g.[7,14] for similar considerations.An extra ingredient is that, following [30], the errors in the symplecticity of P and Lagragianity of L are controlled by the normal component of the invariance error, η N .
The symplectic form on the bundle L, is represented by the the anti-symmetric matrixvalued map Ω L : Td ρ → C n×n , which is where we use the pairwise involution of the first integrals, and the corresponding Hamiltonian vector fields to get Let Ω L : Td ρ → C n×n be the matrix-valued map given by (3.1).Then, Ω L = O n and, in T d ρ , Moreover, for any δ ∈]0, ρ]: Proof.The fact that Ω DK = O d follows directly from the exact symplectic structure, since To do so, we first compute the action of L ω on Ω DK and, using (2.23) and (2.2) we get See [17].Inspired by [30], we obtain formula and symplecticity of ω.To do so using the matrix components, first notice that (3.5) reads for i, j = 1, . . ., d, where the indices r, s, t run in 1, . . ., 2n.Hence, from where we obtain (3.6).Notice also that, since Finally, the quantitive estimate (3.2) follows from Rüssmann and Cauchy estimates from Corollary 2.14 applied to the components of (3.4).In particular, With the previous lemma we control the approximate symplecticity of the frame P .
Lemma 3.2.The matrix-valued map P : Td ρ → C 2n×2n , defined in (2.8), is approximately symplectic, i.e., the simplecticity error map is small in the sense that, for any δ ∈]0, ρ]: Proof.To characterize the error in the symplectic character of the frame, we compute from which the result follows immediately.

Relations between the invariance error and their tangent and normal components.
From the definitions of η L and η N , we obtain easily their bounds controlled by E: In other to control E in terms of η L and η N we have to assume the invertibility of the frame P , which is a consequence of the approximate symplecticity, that is controlled in a narrower strip.We obtain the following lemma.Lemma 3.3.Assume that Then, for any δ ∈]0, ρ]: Proof.The hipothesis implies that Ω L ρ−δ < ν < 1 and Ω N ρ−δ < ν < 1, so the matrices (I n + Ω N Ω L ) and (I n + Ω L Ω N ) are invertible and Then, is invertible, and Notice that, since I 2n + Ω -1 0 E sym = Ω -1 0 P ⊤ Ω•K P , then both P and P ⊤ are invertible in Tρ−δ .
From the definition (2.16) of η we obtain , from where we could obtain easily bounds for E ρ−δ and E ⊤ ρ−δ , but it is better to keep track the dependences with respect to η L ρ and η N ρ separately.Since the bounds (3.9) and (3.10) follow.
We want to avoid using extra Cauchy estimates for DE and (DE) ⊤ , and then loose more analyticity strip.To do so, first, since , from where the bounds (3.11) and (3.12) follow.

Control of the action of the Lie operator.
Here we control the action of the operator L ω on K, L, L ⊤ , G L , B, and N , avoiding the dependence of the estimates on ω.Lemma 3.4.Assume the condition that includes the condition (3.8) in Lemma 3.3.Then, for any δ ∈]0, ρ]: Proof.Estimate (3.14) follows from the identity L ω K = E − X h •K, the bound (3.9) and hypothesis (3.13).Now, we consider the objects L ω L and L ω L ⊤ , given by from where (3.15) and (3.16) follow.

Bound (3.17) follows from
Then, from the identity G L B = I n , we obtain from where we get the estimate (3.18).Notice that G L and B are symmetric, so we obtain the same bounds for their transposes.
In order to bound E NN red , we apply the left operator L ω to the identity L ⊤ Ω•K N = −I n , and obtain Then, using the geometric property (2.2), we obtain from which we obtain (3.24). Finally, . from which we obtain (3.25).

3.2.
One step of the iterative procedure.Here we apply one correction of the (modified) quasi-Newton method described in Subsection 2.6 and we obtain quantitative estimates for the new approximately invariant torus and related objects.We set sufficient conditions to preserve the control of the previous estimates.The constants that appear along the proof are collected in Appendix B, Tables 2 and 3.
Lemma 3.6 (The Iterative Lemma).For any δ ∈]0, ρ/3], there exist constants where then we have a new parameterization K ∈ (A(T d ρ−2δ )) 2n , that defines new objects L, B, N and T (obtained replacing K by K in the corresponding definitions) satisfying

Moreover, the tangent and normal components of the new error of invariance
Proof.We divide the proof into several steps.Starting from the initial parameterization, K, we first consider an intermediate parameterization, and then compute the new parameterization as In the following, we will invoke Lemmas 3.3, 3.4 and 3.5, whose condition (3.8) is included into the hypothesis (3.26) (this corresponds to the first term in (3.27)).
Step 1: Control of the intermediate parameterization.We recall that, in (3.44), Hence, from Rüssmann estimates in Lemma 2.14 with bite ρ to R ω η N we obtain and with bite δ we obtain The intermediate torus should be included in the domain U 0 , more specifically K(T d ρ−δ ) ⊂ U 0 .To verify that, notice that where the last inequality follows if As we will see, this condition is implied by the third term in (3.27).Using Rüssmann and Cauchy estimates, see Lemma 2.16, on and, analogously, ( Notice that, in particular, conditions that, as we will see, are implied by the fourth and fith terms in (3.27).
Let L be the L object associated to K. Notice that, from (3.52), Step 2: Computation of the intermediate invariance errors.The error of invariance in the intermediate step is which can be written as from were we obtain the bound from where we obtain that the intermediate normal error where we emphasize that Notice that the intermediate normal error is quadratically small.Quantitatively, Step 3: Control of the new parameterization.The new approximation is where Using Rüssmann estimates we obtain Notice that K( Td ρ−δ ) ⊂ U 0 , and the components of ξ L are in A(T d ρ−2δ ).Hence, the computation of K in (3.60) could be done if ξ L DK ρ−2δ < δ and ξ L Xp ρ−2δ < r, where we recall r is the width of complex 'time-domain" of Φ s .Since ρ < r, these are conditions that are implied by the hypothesis which corresponds to the second term in (3.27).Then, The new approximation should remain in U 0 .In particular, we observe that where the last inequality follows from hypothesis (3.26) (this corresponds to the third term in (3.27)).We emphasize that this control includes the fact that dist( K(T d ρ−δ ), ∂U 0 ) > 0 (see (3.48)).Moreover, we get (3.28) and (3.35) in Lemma 3.6.
By directly applying Cauchy estimates to the first part of (3.63) and bounds (3.49) and (3.50) we get that correspond to (3.36) and (3.37).Then (3.29) and (3.30) follow from hypotheses which corresponds to the fourth and fifth terms in (3.27), and imply conditions (3.52) and (3.53).
In the following, we write L, N , . . .for the corresponding L, N, . . .objects associated to K. In particular, the new tangent frame is Moreover, since The new restricted metric is and, then, We know that G L is invertible (in Td ρ ), and B = G -1 L with B ρ < σ B .We introduce now the constants that corresponds to the sixth term in (3.27), from Lemma A.1 we obtain that GL is invertible (in Td ρ−3δ ) and B − B from where we obtain that We obtain estimates (3.31) and (3.38) in Lemma 3.6.
The new normal frame is Then, (3.32) and (3.33) follow from hypotheses which corresponds to the seventh and eighth terms in (3.27).Hence, we obtain estimates Moreover, since we obtain We know that T is invertible and Mimicking the arguments made above for G L and B = G -1 L , we introduce the constants which corresponds to the ninth term in (3.27), we obtain that T is invertible and from where we obtain that We obtain then the estimates (3.34) and (3.41) on the new object.
Step 4: Computation of the new invariance errors.In order to compute the new invariance error Ē, we first compute then (3.76) In oder to get a crude bound of Ē from the first line in (3.76), we first bound , and, then, Ē (3.78) A posteriori, we will see that Ē is quadratically small.To compute the new normal error ηN we first compute the new tangent frame L. To do so, we first obtain and, using (3.75), we get We observe that ηN is quadratically small.Quantitatively, since where we use condition (3.62), were we apply Lemma 3.1 to K and bound (3.59), and, finally, applying (3.77) and hypothesis 3.26, coming from the tenth term of (3.27), we get which corresponds to the bound (3.42) in Lemma 3.6.
The new tangent error is where the addends are numbered in order.In the previous expression, all addends, but the first, are trivially quadratically small.But, in fact, L ω ξ L is selected in such a way that ηL , where we apply that N ⊤ Ω•K L = I n , (3.57), and L ω ξ L = η L − T ξ N .From this, we get a bound for the first addend of (3.81): (3.82) ηL For the second addend, since we get (3.83)ηL applying Cauchy estimates.The third addend is L ω ξ L , from which, proceeding analogously, we obtain the bound ηL In order to bound the fourth addend, we first observe that which is bound (3.43).With this we finish the proof of Lemma 3.6.

3.3.
Convergence of the iterative process.Once the quadratic procedure has been established in Section 3.2, proving the convergence of the scheme follows standard arguments, that we will detail for providing explicit conditions for the KAM theorem.
Proof of Theorem 2.18.Let us consider the parameterization K 0 := K with initial invariance error E 0 := E, whose tangent and normal projections are η L 0 and η N 0 , respectively.We also introduce L 0 := L, L ⊤ 0 := L ⊤ , B 0 := B, N 0 := N , N ⊤ 0 := N ⊤ , and T 0 := T associated to the initial parameterization.By applying Lemma 3.6 recursively, at the step j we obtain new objects K , and T j := T j−1 .The domain of analyticity of these objects is reduced at every step, from the initial value ρ 0 = ρ to a limiting value ρ ∞ .At the step j, the parameterization K j and associated objects are defined in a strip of width ρ j , and have been produced from the parameterization K j−1 , which is defined in a strip of width ρ j−1 , throughout computations (involving small divisors equations and derivatives) that produce three bites of size δ j−1 to the width ρ j−1 , so then If we select a geometric sequence of bites δ j = δ 0 a j with a > 1, then, from the identity Let us assume that we have successfully applied j times Lemma 3.6 We observe that condition (3.26) is required at every step, but the construction has been performed in such a way that we control dist(K j (T d ρj ), ∂U 0 ), L j := Lj−1 , L ⊤ j := L⊤ j−1 , B j := Bj−1 , N j := N j−1 , N ⊤ j := N⊤ j−1 , and T j -1 uniformly with respect to j, so the constants that appear in Subsection 3.1, displayed in Table (1), and in Lemma 3.6, displayed in Tables 2 and 3, are taken to be the same for all steps by considering the worst value of δ j , that is, δ 0 .We also take the upper bounds ĉR and ĉ1 R for c R (δ j ) and ĉ1 R (δ j ).The first computation is tracking the sequence ε j of errors, defined to be By defining where we used the sums 1+2+. ..+2 j−1 = 2 j −1, and 1(j−1)+2(j−2)+2 2 (j−3) . ..+2 j−2 1 = 2 j − j − 1.
By imposing which is included in (2.24), we have Now, using expression (3.92), we check Hypothesis (3.26) of the iterative lemma, Lemma 3.6, so that we can perform the step j + 1.The required sufficient condition is included in the hypothesis (2.24) of the KAM theorem, whose inequality has several terms that correspond to the different components in (3.27).
The first condition, using (3.92), is given by where the last inequality is included in (2.24).Checking the second conditions is analogous, and it is where the last inequality is again included in (2.24).Also the last, since: which is also included in (2.24).
In order to check the rest of conditions for K j , L j , L ⊤ j , B j , N j , N ⊤ j and T j -1 , we have to relate them to the conditions corresponding to the initial objects K 0 , L 0 , L ⊤ 0 , B 0 , N 0 , N ⊤ 0 and T 0 -1 .The third condition in (3.26) is checked as follows.We recursively obtain that dist which is included as a condition into (2.24).
In order to check the fourth condition, we again proceed recursively to obtain that and then we include the last inequality as ).The rest of conditions in (3.26), associated to L ⊤ j , B j , N j , N ⊤ j and T j -1 , follow by reproducing the same computations.
and then we include the last inequality as ).The rest of conditions in (3.26), associated to (DK) ⊤ j , B j , N j , N ⊤ j and T j -1 , mutatis mutandis.
Remark 3.8.The choice of the δ j above as the geometric series with ratio 1/2 is justified by the following rationale.Let us assume that the constants involved in the theorem do not depend on δ (their dependence on it is very mild) then, 3.90 can be written as where δ j > 0 for all j ≥ 0.Then, if (δ 0 j = 2 −j−1 ∆) j≥0 is the target geometric sequence obtained in Remark 3.7, for any other positive sequence (δ j ) j≥0 satisfying (3.102), the function f : [0, 1] → R defined as is strictly convex and f ′ (0) = 0. 3) (3.12) κ , a τ +1 Qη κ (3.99) and D r f U = ℓ1,...,ℓr ∂ r f ∂x ℓ1 . . .∂x ℓr U , that could be infinite.For M : U ⊂ C m → C n1×n2 , we consider the norms

Theorem 2 . 18 ( 1
KAM theorem with first integrals).Let (ω, g, J ) be a compatible triple on the open set U ⊂ R 2n , where the symplectic form is exact: ω = dα.Let h : U → R be a Hamiltonian for which there exists a moment map p : U → R n−d whose components and h are pairwise in involution functionally independent functions, being Φ : D ⊂ R n−d × U → U the corresponding moment flow.Let ω ∈ D d γ,τ be a Diophantine vector, for some constants γ > 0 and τ ≥ d − 1.Let K : T d → U be a parameterization.We assume that the following hypotheses hold.H The geometric objects ω, g, J , α, the Hamiltonian h, and the moment map p can be analytically extended to an open complex set U ⊂ C 2n covering U , and the moment flow to an open complex set D ⊂ C n−d × U covering D.Moreover, there exist constants c

Table 3 :
Constants in the proof of Lemma 3.6, steps 3 and 4.