Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 126, pp. 1–26. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF KAM TORI FOR PRESYMPLECTIC VECTOR FIELDS SEAN BAUER, NIKOLA P. PETROV Abstract. We prove the existence of a torus that is invariant with respect to the flow of a vector field that preserves the presymplectic form in an exact presymplectic manifold. The flow on this invariant torus is conjugate to a linear flow on a torus with a Diophantine velocity vector. The proof has an “a posteriori” format, the the invariant torus is constructed by using a Newton method in a space of functions, starting from a torus that is approximately invariant. The geometry of the problem plays a major role in the construction by allowing us to construct a special adapted basis in which the equations that need to be solved in each step of the iteration have a simple structure. In contrast to the classical methods of proof, this method does not assume that the system is close to integrable, and does not rely on using action-angle variables. 1. Introduction The goal of this article is to give a proof of the existence of a torus that is invariant with respect to the flow of a presymplectic vector field V in an exact presymplectic manifold (P,Ω), i.e., in a manifold P endowed with an exact constant-rank 2-form Ω that is preserved under the flow of V . Perhaps the most prominent occurrence of presymplectic manifolds in physics is in the geometric theory of dynamical systems with constraints. These are systems for which the transition from Lagrangian to Hamiltonian description is non-trivial because some of the relations pA := ∂L ∂q̇A (q, q̇) expressing the generalized momenta pA in terms of generalized velocities q̇A cannot be solved for q̇A since the matrix ( ∂2L ∂q̇A ∂q̇B ) is degenerate; the relations that cannot be solved play the role of con- straints. The modern theory of constrained systems was initiated in the early 1950s by Dirac [23, 24, 25] and developed by Bergmann and his collaborators for purposes of quantization of field theories [4, 10, 44] (the book [52] offers an in-depth exposi- tion). Such situations occur in also classical electromagnetic theory [53, Ch. V], in the description of relativistic particles [36], [53, Ch. VII], gauge fields [38, 47, 53]. A geometric theory of constrained systems was proposed in the late 1970s by Gotay and collaborators [33, 34, 35]. In their approach, the system is transformed in stages, and the process ends up with a manifold that is typically presymplectic. Presymplectic geometry is also related to equivalence between Lagrangian and Hamiltonian formalisms for constrained systems [7, 8, 15], geometric approach to 2010 Mathematics Subject Classification. 34D35, 37J40, 70K43, 70H08. Key words and phrases. KAM theory; invariant torus; presymplectic manifold; stability. c©2020 Texas State University. Submitted April 5, 2019. Published December 22, 2020. 1 2 S. BAUER, N. P. PETROV EJDE-2020/126 maximum principles [6], geometric optics [17, 18, 27], etc. Other topics of inter- est are canonical transformations in presymplectic systems [14, 16], reduction of presymplectic manifolds [2, 22, 28, 29, 45], etc. A major achievement in the theory of Hamiltonian systems in the second half of the XX century was the celebrated Kolmogorov-Arnold-Moser (KAM) Theorem. This theorem is the subject of multiple reviews and pedagogical expositions (see, e.g., [5, 11, 19, 20, 41, 46, 50, 51]), and its history is beautifully described in the recent book by Dumas [26]. The standard proofs of the KAM-type theorems perform an infinite sequence of canonical transformations to convert a slightly perturbed integrable system on a symplectic manifold into action-angle variables. González, Jorba, de la Llave and Villanueva developed a new method of proof in their seminal 2005 paper [21]. This method relies heavily on the geometry of the system. One important ingredient in their proof is the so-called automatic reducibility: if K is a torus in the symplectic manifolds P that is invariant with respect to a map f : P → P, then the tangent bundle to K is preserved under the derivative Tf . This simplifies the structure of the coefficient matrices in certain difference equations which, in turn, dramatically simplifies the solution of the problem. Methods similar to the ones developed in [21] have since been used in [32] to study the existence of non-twist tori in degenerate Hamiltonian systems, and in [30, 42] to prove the existence of lower dimensional invariant tori that are partially hyperbolic or elliptic. Since these methods are suitable for efficient numerical implementation, they have been used for this purpose in [12, 13, 31, 39]. Many aspects of these methods are considered in the recent book by Haro et al [37] (KAM theory is the subject of Chapter 4). Alishah and de la Llave [3] used the ideas of [21] to prove a KAM theorem for presymplectic systems, for which the degeneracy of the presymplectic form complicates the matters. They considered a family {fλ} of maps that preserve the presymplectic form, and found a value λ̄ of the parameter λ and an embedding K from a torus to the presymplectic manifold such that fλ̄ ◦ K = K ◦ Tω where Tω : θ 7→ θ + ω is translation on the torus by a Diophantine vector ω. The main goal of this paper is to prove a KAM theorem for a family {Vλ} of presymplectic vector fields on an exact presymplectic manifold (P,Ω), with dimP = d + 2n, dim ker Ω = d, Ω = dτ for some τ ∈ Ω1(P). For most of the paper we consider P ∼= Td × T ∗Tn ∼= Td+n × Rn, where ker Ω coincides with the first d dimensions. Our goal is to find a value λ̄ of the parameter λ ∈ Rd+2n and an embedding K : Td+n → P such that the submanifold K := K(Td+n) is invariant with respect to the flow Φλ̄,t of the vector field Vλ̄, and K conjugates Φλ̄,t to the linear flow φt : Td+n → Td+n : θ 7→ θ+ tω, where ω ∈ Rd is a constant Diophantine vector: Φλ̄,t ◦K = K ◦ φt , t ≥ 0 . (1.1) The infinitesimal form of (1.1) is Vλ̄,K(θ) = K∗θ ωθ, where K∗θ : TθTd+n → TK(θ)P is the derivative of K at θ ∈ Td+n and we consider ω ∈ Rd+n as ωθ ∈ TθTd+n = Rd+n. Our proof of the theorem has an a posteriori format (as in [21]). In more detail, we assume the existence of λ0 and K0 : Td+n → P that satisfy (1.1) only approximately, i.e., Φλ0,t ◦K0 ≈ K0 ◦ φt. Then we start a version of the Newton EJDE-2020/126 KAM TORI FOR PRESYMPLECTIC VECTOR FIELDS 3 method to construct iteratively a sequence of better and better approximations (λ0,K0) 7→ (λ1,K1) 7→ (λ2,K2) 7→ (λ3,K3) 7→ · · · (1.2) whose limit (λ∞,K∞) is the desired solution (λ̄,K) satisfying (1.1). The loss of domain accompanying each step of the iteration is compensated by the fast convergence of the Newton method. This method is convenient to implement in numerical computations. Moreover, a posteriori theorems are suitable for validation of numerical results, i.e., they can be used to produce computer assisted proofs of existence of invariant manifolds. In this article we extensively use the geometry of the system to our advantage, inspired by the ideas of [21, 3]. If K = K(Td+n) is the invariant torus, then at each point k ∈ K the kernel ker Ωk of the presymplectic form is a subspace of TkK, so that we have ker Ωp ⊆ TkK ⊆ TkP. In fact, we have much more – a filtration of subbundles ker Ω ⊆ TK ⊆ TP|K which is invariant with respect to the flow Φλ̄,t of the vector field Vλ̄. This and the invariance (1.1) allow us to construct a special basis adapted to the filtration, in which the matrices of the operators have zero blocks. Even if the torus is only approximately invariant (as in the case of (λj ,Kj)), these blocks, albeit non-zero, are small and we have good bounds on their norms. An important role is also played by the fact that K is isotropic (i.e., that the the pull-back of Ω to K vanishes identically), and the approximately invariant tori are approximately isotropic. We found the following interesting quotations related to this fact. On page 45 of his classic 1973 monograph [43], Moser writes Actually, more than asserted in Theorem 2.7 can be proven. It turns out that the differential form ∑n k=1 dyk ∧ dxk vanishes identically on the tori (3.11), and one calls manifolds with this property and of maximal dimension Lagrange manifolds. In this quotation, Theorem 2.7 is (as Moser calls it) the Kolmogorov-Arnold The- orem, and the tori (3.11) are the invariant tori whose existence is proved in the KAM theorem. On page 584 of their monumental book [1], Abraham and Marsden write Moser [1973a] states that the invariant tori are Lagrangian sub- manifolds [. . .]. This fact can probably be exploited, although to our knowledge it has not been. The fact that K is isotropic and of a maximum dimension (i.e., Lagrangian in the symplectic case) is crucial for the proofs in [21, 3] and in this paper. The paper is organized as follows. In Sections 2.1–2.5 we introduce some basic definitions and notations, discuss the integrability of the distribution ker Ω and con- struct the symplectic manifold P/ ker Ω. The main theorem is stated in Section 2.6. In Section 3 we study the geometric structures occurring when we know the true solution (λ̄,K) of the problem – we prove that K is isotropic (Section 3.1), give a detailed construction of the basis adapted to the filtration ker Ω ⊆ TK ⊆ TP|K (Section 3.2), and study the properties of the matrix of transition from a general basis of TP|K to the special adapted basis (Section 3.3). Section 4 is devoted to the properties of approximate solutions. Approximately isotropic tori are studied in Section 4.1. In Section 4.2 we derive an equation for the corrections εj and ∆j needed to obtain a better approximation λj+1 = λj + εj , Kj+1 = Kj + ∆j . We solve this equation in Section 4.3, relying heavily 4 S. BAUER, N. P. PETROV EJDE-2020/126 on the machinery developed in Section 3. Finally, in Section 5, we collect several lemmata that justify the applicability of the Newton method for performing the iteration (1.2). 2. Preliminaries and general setup In this section we set up the problem and state our main result. 2.1. Presymplectic manifolds and vector fields. Definition 2.1. A presymplectic manifold is a pair (P,Ω), where P is a manifold of any (finite) dimension and Ω ∈ Ω2(P) is a closed 2-form with constant rank. If Ω is exact, i.e., if Ω = dτ for some τ ∈ Ω1(P), then we say that (P,Ω) is an exact presymplectic manifold. Throughout this paper, we will always assume that dimP = d+ 2n , rank Ω = 2n . (2.1) Most of the time we will consider the specific exact presymplectic manifold P := Td × T ∗Tn ∼= Td × Tn × Rn (2.2) with an exact presymplectic form Ω with ker Ω = Td. We assume that Td×T ∗Tn is endowed with an Euclidean structure, so that we can identify two-forms with linear operators and abstract tangent vectors with column vectors. In the definition below, X(P) stands for the vector fields on P, L is the Lie derivative, and ι is the interior product, i.e., the contraction with a vector field. Definition 2.2. Let V ∈ X(P) and Φt : P → P be the time-t flow of V . The vector field V is said to be presymplectic if Φ∗t Ω = Ω for all t ∈ R. Lemma 2.3. Let (P,Ω) be a presymplectic manifold, and V ∈ X(P). Then the following conditions are equivalent: (a) V is a presymplectic vector field; (b) the Lie derivative of the presymplectic form along V vanishes: LV Ω = 0; (c) the 1-form ιV Ω is closed. Proof. The equivalence of (a) and (b) comes directly from the definition of a Lie derivative, and the equivalence of (b) and (c) follows from Cartan’s magic formula and the closedness of Ω: 0 = LV Ω = ιV dΩ + d(ιVΩ) = d(ιVΩ). � 2.2. Foliation induced by ker Ω. In this section we discuss some results about a general presymplectic manifold (P,Ω) (not necessarily (2.2)). For any p ∈ P, define ker Ωp := {Wp ∈ TpP : ιWpΩp = 0} = {Wp ∈ TpP : Ωp(Wp, Up) = 0 ∀Up ∈ TpP} ⊆ TpP . The subspaces ker Ωp form a differentiable distribution, ker Ω, of rank d. Define Xker Ω(P) := {W ∈ X(P) : ιWΩ = 0} = {W ∈ X(P) : Wp ∈ ker Ωp ∀p ∈ P} . Using the classical Frobenius Theorem (see, e.g., [48, Section 3.5], or, especially, [40, Appendix 3]) and the fact that Ω is closed, one can easily obtain Lemma 2.4. If Ω is presymplectic, the distribution ker Ω is integrable. EJDE-2020/126 KAM TORI FOR PRESYMPLECTIC VECTOR FIELDS 5 Lemma 2.4 implies that P has a foliation with d-dimensional leaves such that the tangent space to the leaf through p ∈ P at p is ker Ωp. We make the additional assumption that the collection of leaves of the foliation forms a smooth manifold Q (for a discussion see, e.g., [40, Sec. 4.3.3 of Appendix 3]). Let πQ : P → Q be the canonical projection taking each point p ∈ P to the leaf (πQ)−1(πQ(p)) through it. If πQ∗ : TP → TQ is the derivative of πQ, then kerπQ∗ = ker Ω. The lemma below (see [40, Section III.7]) states that Q carries a natural symplectic structure. Lemma 2.5. Let (P,Ω) be a presymplectic manifold such that ker Ω determines an integrable foliation of P whose leaves form a smooth manifold Q, and let πQ : P → Q be the canonical projection. Then there exists a unique symplectic form Ω̃ ∈ Ω2(Q) on the manifold Q such that (πQ)∗Ω̃ = Ω. For the case (2.2) considered in this paper, Q ∼= T ∗Rn. It would be interesting to investigate the case when P has a more complicated structure than (2.2). 2.3. Matrix representation of Ω and Ω̃. We consider the case when P has product structure (2.2), so ker Ωp ∼= Td for every p ∈ P and the collection of leaves, Q = P/ ker Ω = T ∗Tn , (2.3) is a symplectic manifold with symplectic form Ω̃. Because of the assumed Euclidean structure on P, we can identify a 2-form on P with a linear operator. For any p ∈ P, let Jp : TpP → TpP be the linear operator corresponding to Ωp, defined by 〈Up, JpWp〉Rd+2n := Ωp(Up,Wp) , Up,Wp ∈ TpP ∼= Rd+2n , (2.4) where 〈·, ·〉Rd+2n is the Euclidean inner product on Rd+2n. Similarly, for any q ∈ Q, let J̃q : TqQ → TqQ be the linear operator corresponding to the symplectic form Ω̃q on TqQ at q ∈ Q: if 〈·, ·〉R2n is the Euclidean inner product on R2n, then〈 Ũq, J̃qW̃q 〉 R2n := Ω̃q ( Ũq, W̃q ) , Ũq, W̃q ∈ TqQ ∼= R2n . (2.5) If we choose a basis for TpP ∼= Rd×R2n such that the first d vectors form a basis of Rd = Tp ker Ω, and the other 2n vectors form a basis of R2n ∼= TpT ∗Tn, then we can write Jp in a matrix form as Jp = [ 0 0 0 J̃πQ(p) ] , J>p = −Jp , J̃ >q = −J̃q . (2.6) Although Jp is not invertible, we define J−1 p as the Moore-Penrose pseudoinverse: J−1 p := [ 0 0 0 J̃−1 πQ(p) ] , so that Jp J −1 p = [ 0 0 0 I2n ] . (2.7) 2.4. Miscellaneous definitions and results. Definition 2.6. For γ > 0 and σ ≥ d+ n− 1, the set of all ω ∈ Rd+n satisfying |ω · k| ≥ γ |k|σ ∀k ∈ Zd+n \ {0} (2.8) is called the set of Diophantine vectors and is denoted by D(γ, σ). 6 S. BAUER, N. P. PETROV EJDE-2020/126 For any ρ > 0, we define the torus “thickened” into the complex direction, Td+n ρ := {θ ∈ Cd+n/Zd+n : | Im θα| ≤ ρ, α = 1, 2, . . . , d+ n} . (2.9) Let | · | stand for the supremum norm on Rm or Cm (for any m). Given ρ > 0, we define the set of functions Wρ as follows: Wρ := { K : Td+n ρ → P :(a)K is real analytic on the interior of Td+n ρ , (b)K is continuous on the boundary of Td+n ρ , (c)K is periodic of period 1 in all arguments } . (2.10) Define ‖K‖ρ = supθ∈Td+nρ |K(θ)|, so that (Wρ , ‖·‖ρ) is a Banach space. For analytic functions g : B → C (where B ⊆ C), and any ` ∈ B, define the norms |g|C`,B := sup 0≤|k|≤` sup z∈B ∣∣Dkg(z) ∣∣ . (2.11) Lemma 2.7 (Cauchy bound). For K ∈ Wρ and 0 < δ < ρ, ‖DK‖ρ−δ ≤ Cδ−1‖K‖ρ . (2.12) Lemma 2.8 (Rüssmann [49]). Let ω = [ω1 ω2 · · · ωd+n]> ∈ D(γ, σ) and let the function h : Td+n → P be analytic on Td+n ρ and have zero average. Then for any 0 < δ < ρ, the differential equation ∂ωv = h, where ∂ω := ∑d+n α=1 ω α ∂ ∂θα is the directional derivative in the direction of ω, has a unique average-zero solution v : Td+n → P which is analytic in Td+n ρ−δ . The solution v satisfies the estimate ‖v‖ρ−δ < Cγ−1δ−σ‖h‖ρ , (2.13) where C is a constant depending only on d, n, and σ. 2.5. General setup and matrix notation. Let {Vλ} be a (d + 2n)-parameter family of presymplectic vector fields on the exact presymplectic manifold (P,Ω). Our goal is to construct a smooth embedding K : Td+n → P (2.14) such that K := K(Td+n) be invariant with respect to the flow Φλ̄,t of Vλ̄ for some value λ̄ and the flow Φλ̄,t on K be conjugate to the linear flow φt on Td+n: Φλ̄,t(K(θ)) = K(φt(θ)) ∀t ∈ R , ∀θ ∈ Td+n , (2.15) where ω ∈ Rd+n is a constant Diophantine vector and φt : Td+n → Td+n : θ 7→ θ + tω. (For elements of Td+n, e.g., θ + tω, we assume that we take the fractional part of each component.) Differentiate (2.15) with respect to t and set t = 0 to obtain Vλ̄,K(θ) = K∗θωθ ∀θ ∈ Td+n . (2.16) Here Vλ̄,K(θ) ∈ TK(θ)K ⊆ TK(θ)P is the value of Vλ̄ at K(θ) ∈ K, ωθ ∈ TθTd+n is the Diophantine vector ω considered as an element of TθTd+n = Rd+n, and K∗θ : TθTd+n → TK(θ)K ⊆ TK(θ)P is the derivative of K at θ. Instead of the differential-geometric notations used in (2.16), we will normally use matrix notations. In these notations (2.16) reads Vλ̄,K(θ) = DKθ ω or Vλ̄,K(θ) = ∂ωKθ , (2.17) EJDE-2020/126 KAM TORI FOR PRESYMPLECTIC VECTOR FIELDS 7 where ω is considered as a constant column vector, ∂ω := ω · ∇, and DKθ := [ (DKθ) A α ] = [∂KA ∂θα (θ) ] ∈ Md+2n,d+n(R) . Throughout the paper we will systematically use the notations collected in Table 1. Table 1. Notation for indices and coordinates Coordinates Range of indices Description x = (xA A,B = 1, 2, . . . , d+ 2n Coordinates in P ∼= Td × T ∗Tn x = (xµ) µ, ν = 1, 2, . . . , d Coordinates in Td (the first d in P) x̃ = (x̃i) i, j = 1, 2, . . . , 2n Coordinates in T ∗Tn (the last 2n in P) θ = (θα) α, β = 1, 2, . . . , d+ n Coordinates in Td+n 2.6. Statement of the main theorem. Theorem 2.9. Assume that: (1) ω ∈ D(γ, σ) is a Diophantine vector; (2) P = Td × T ∗Tn; (3) Ω is an exact presymplectic form on P of rank 2n such that the kernel of Ω coincides with the first d directions; (4) {Vλ} is a (d+ 2n)-parameter family of analytic presymplectic vector fields on P; (5) K0 : Td+n → P is an embedding belonging to the class Wρ0 (2.10); (6) the value λ0 of the parameter λ is such that the pair (λ0,K0) is non- degenerate in the sense of Definition 4.7; (7) each vector field from the family {Vλ} can be holomorphically extended to some complex neighborhood Br of K0(Td+n ρ ), where Br := {z ∈ Cd+2n | ∃θ ∈ Td+n ρ0 such that |z −K0(θ)| < r} , (2.18) for some r > 0 and such that |Vλ|C2,Br is finite. We define the error function e0,θ := Vλ,K0(θ)−∂ωK0,θ. Then there exists a constant c > 0 which depends on d, n, σ, ρ0, ‖DK0‖ρ0 , r, |Vλ|C2,Br , ∥∥∂Vλ ∂λ ∣∣ λ=λ0 ◦ K0‖ρ0 , and ∣∣{avg (Λ0)}−1∣∣ (see (4.32)), such that if 0 < δ0 < max{1, ρ012} and the error e0 satisfies ‖e0‖ρ0 ≤ min{γ4δ4σ 0 , crγ2δ2σ 0 ‖e0‖ρ0} , then there exists a mapping K ∈ Wρ0−6δ0 and a vector λ̄ ∈ Rd+2n such that (2.16) (or, equivalently, (2.17)) is satisfied. Moreover, the following inequalities hold: ‖K −K0‖ρ0−6δ0 < 1 c γ2δ−2σ 0 ‖e0‖ρ0 , |λ̄− λ0| < 1 c γ2δ−2σ 0 ‖e0‖ρ0 . 3. Exact solutions In this section we will assume that we know the exact solution of the problem and will use the geometry of the problem to construct bases of TK with special properties that will be utilized in Section 4. 8 S. BAUER, N. P. PETROV EJDE-2020/126 3.1. Invariant tori are isotropic. Definition 3.1. Let {Vλ} be a (d+ 2n)-parameter family of presymplectic vector fields on the exact presymplectic manifold (P,Ω), and ω ∈ Rd+n be a Diophantine vector. If for some value λ̄ of λ there exists an embedding K : Td+n → P such that (2.17) holds, we call K and K an invariant torus (KAM torus) or a true solution. This terminology is somewhat of a misnomer. We require more than K being merely invariant under the flow of Vλ̄ – we want that the motion on K be quasi- periodic, (i.e., we require that the dynamics on K be conjugate to a linear flow on Td+n with frequency ω independent over the rationals). Notational convention. In Section 3 we assume that λ = λ̄, and set V := Vλ̄. Definition 3.2. An invariant (in the sense of Definition 3.1) torus, K = K(Td+n), in the presymplectic manifold (P,Ω) is said to be isotropic if the pull-back, K∗Ω ∈ Ω2(Td+n), of Ω ∈ Ω2(P) to the torus Td+n vanishes identically. In Lemma 3.3 below we will prove that an invariant torus is isotropic. Similar results for maps are well-known for the case of submanifolds invariant with respect to symplectic or presymplectic maps (see, e.g., [21, Section 4, Lemma 1] or [3, Lemma 2.5]). This fact is crucial in the proof of Lemma 4.2 which, in turn, is essential for the bounds needed to solve the linearized equation in Section 4.3. We introduce the linear operator Lθ : TθTd+n → TθTd+n as the matrix repre- sentation of the pull-back (K∗Ω)θ: for Uθ,Wθ ∈ TθTd+n, 〈Uθ, LθWθ〉Rd+n := (K∗Ω)θ (Uθ,Wθ) = ΩK(θ) (K∗θ Uθ,K∗θWθ) . (3.1) The explicit expression for the matrix elements of Lθ is Lθ = DK>θ JK(θ)DKθ ∈ Md+n,d+n(R) . (3.2) Lemma 3.3. Let (P,Ω) be an exact presymplectic manifold, V ∈ X(P) be presym- plectic, and K : Td+n → P be a true solution. Then the invariant torus K = K(Td+n) is isotropic (i.e., K∗Ω and, hence, Lθ, vanish identically). Proof. We prove the lemma in two steps: first we use the exactness of Ω to show that the average (over Td+n) of each matrix element of L is zero, and then we use the ergodicity of the flow θ 7→ θ + tω on Td+n to demonstrate that K∗Ω and, therefore, L, are constant on Td+n. Since the presymplectic form Ω is exact, there exists a 1-form τ ∈ Ω1(P) such that Ω = dτ , hence K∗Ω = K∗(dτ) = d(K∗τ). If τK(θ) = d+2n∑ A=1 τA(K(θ)) dxA , then the pull-back K∗τ ∈ Ω1(Td+n) is given by (K∗τ)θ = d+n∑ α=1 Cα(θ) dθα , Cα(θ) := d+2n∑ A=1 τA(K(θ)) ∂KA ∂θα (θ) , and the matrix representation of (K∗Ω)θ is( Lθ )α β = ∂Cα ∂θβ (θ)− ∂Cβ ∂θα (θ) . EJDE-2020/126 KAM TORI FOR PRESYMPLECTIC VECTOR FIELDS 9 Because of the periodicity of the functions Cα : Td+n → R, avg (∂Cα ∂θβ ) := ∫ Td+n ∂Cα ∂θβ (θ) dθ1 dθ2 · · · dθd+n = ∫ Td+n−1 (∫ T1 ∂Cα ∂θβ (θ) dθβ ) dθ1 · · · d̂θβ · · · dθd+n = 0 (3.3) (the term d̂θβ is missing), so avg (L) = 0 and, hence, avg (K∗Ω) = 0. Now we prove that L and K∗Ω are constant on Td+n. Restrict the target space of the map K (2.14) from P to the image K of K, to obtain the diffeomorphism Kr := K|Td+n→K : Td+n → K. Since K is invariant with respect to the flow of V ∈ X(P), at each k ∈ K, Vk ∈ TkK. Therefore, the restriction V |K of V to K can be considered as a vector field on K: V |K ∈ X(K). For the same reason, the Lie derivative with respect to V has a natural restriction to sections of any tensor power of the tangent and cotangent bundles of K. The pull-back of V |K by the diffeomorphism Kr is K∗r V := ( K−1 r ) ∗ (V |K) ∈ X(Rd+n). If we consider the constant ω ∈ Rd+n as a tangent vector ωθ ∈ Tθ(Td+n), then the pull-back of VK(θ) = K∗θ ωθ ∈ TK(θ)K is (K∗r V )θ = ( K−1 r ) ∗K(θ) K∗θ ωθ = ( K−1 r ◦K ) ∗θ ωθ = ωθ . By a well-known property of the Lie derivative, K∗LV Ω = LK∗r VK ∗Ω = LωK∗Ω (where all objects and operations are restricted to K). Since V is presymplectic, LV Ω = 0, which implies that its pull-back K∗Ω to Td+n is constant on the orbits of the flow θ 7→ θ + tω, t ∈ R. But ω is Diophantine, hence this flow is ergodic on Td+n, therefore K∗Ω = const and L = const. This together with the fact that avg (L) = 0 and avg (K∗Ω) = 0 implies the desired result. � 3.2. Construction and properties of an adapted basis of TK(θ)P. In this section we construct a basis of (TP)|K that is adapted to the invariant (with respect to the flow of the presymplectic vector field V ) filtration (ker Ω)|K ⊆ TK ⊆ (TP)|K of vector bundles over K. 3.2.1. Adapted coordinates in Td+n and a basis of TK(θ)K. We first construct a basis of TK(θ)K (at an arbitrary point K(θ) ∈ K) as a push-forward { K∗θ ( ∂ ∂θα ) θ }d+n α=1 of the basis { ( ∂ ∂θα ) θ }d+n α=1 of TθTd+n. Every vector in TK(θ)K has the form K∗θUθ for some Uθ ∈ TθTd+n. The com- ponents (K∗θUθ) A of K∗θ Uθ in the basis {( ∂ ∂xA ) K(θ) }d+2n A=1 are related to the com- ponents Uαθ of Uθ in the basis { ( ∂ ∂θα ) θ }d+n α=1 by (K∗θ Uθ) A = d+n∑ α=1 ∂KA ∂θα (θ)Uαθ . (3.4) If we think of (K∗θ Uθ) A as a column vector K∗θUθ = [ (K∗θUθ) 1 · · · (K∗θUθ)d+2n ]> and of the (d+ n) columns of the matrix DKθ = [∂KA ∂θα (θ) ] = [∂K ∂θ1 (θ) · · · ∂K ∂θd+n (θ) ] ∈ Md+2n,d+n(R) , (3.5) 10 S. BAUER, N. P. PETROV EJDE-2020/126 as (d + 2n)-dimensional vectors, then we can interpret (3.4) as expressing the ar- bitrary vector K∗θ Uθ ∈ TK(θ)K as a linear combination of ∂K ∂θ1 (θ), . . . , ∂K ∂θd+n (θ). Therefore, the column vectors { ∂K ∂θα (θ) }d+n α=1 form a basis of TK(θ)K. To adapt the basis { ∂K∂θα (θ)}d+n α=1 to the special subspace ker ΩK(θ) ⊆ TK(θ)K, recall that the first d coordinates in P = Td × T ∗Tn (2.2) correspond to ker Ω. Because of this, we reorder the coordinates θα, α = 1, . . . , d+ n, of Td+n in such a way that the d× d block in the upper left corner of DKθ (3.5) has full rank (then the 2n× n block in the lower right corner of DKθ will also be of full rank). Then{ ∂K ∂θµ (θ) }d µ=1 is a basis of ker ΩK(θ), while the vectors ∂K ∂θd+1 (θ), . . . , ∂K ∂θd+n (θ) span an n-dimensional subspace that is transversal to ker ΩK(θ) in TK(θ)K. Define the matrices Zθ ∈ Md+2n,d(R) and Xθ ∈ Md+2n,n(R) as the first d, respectively the last n, columns of DKθ (3.5), so that DKθ = [ [Zθ Xθ ] . Recall the notation from Table 1 for the coordinates x = (xA) = ( x, x̃ ) , in P (2.2): x = (xµ) = ( x1, . . . xd ) , x̃ = (x̃i) = ( x̃ 1, . . . , x̃ 2n ) = ( xd+1, . . . xd+2n ) . We will use these notation in matrices with d + 2n rows – the underscore for the first d rows, and the tilde for the remaining 2n rows: Zθ = [ Zθ Z̃θ ] , Xθ = [ Xθ X̃θ ] , DKθ = [ Zθ Xθ ] = [ Zθ Xθ Z̃θ X̃θ ] . (3.6) 3.2.2. Adapted basis of TK(θ)P. Having constructed a basis of TK(θ)K, we need n more vectors that span the complement of TK(θ)K in TK(θ)P. We will construct them in such a way that, together with the columns ∂K ∂θd+1 (θ), . . ., ∂K ∂θd+n (θ) of Xθ, they form a symplectic basis of TK(θ)Q (2.3). To this end we will use the matrix representation J̃K(θ) (2.5) of the symplectic form Ω̃ on Q, as well as the Gramian matrix of the vectors ∂K ∂θd+1 (θ), . . . , ∂K ∂θd+n (θ), i.e., the matrix X>θ Xθ ∈ Mn,n(R) of their inner products with respect to the Euclidean inner product on T ∗Tn. Define Rθ := ( X̃>θ X̃θ )−1 ∈ Mn,n(R) , (3.7) Ỹθ := J̃−1 K(θ) X̃θ Rθ ∈ M2n,n(R) , (3.8) Yθ := [ 0 Ỹθ ] = [ 0 J̃−1 K(θ)X̃θRθ ] ∈ Md+2n,n(R) . (3.9) Since J̃−1 K(θ), X̃θ, and Rθ are of maximal rank, Ỹθ and Yθ are of maximal rank: rank Ỹθ = rank Yθ = n. We think of the n columns of Yθ as of as vectors in TK(θ)P. Let Zθ,µ, Xθ,a, Yθ,a, with µ = 1, . . . , d, a = 1, . . . , n, stand for the columns of Zθ, Xθ, and Yθ. These d+ 2n vectors are a basis of TK(θ)P with the properties span{Zθ,µ}dµ=1 = ker ΩK(θ) , span {{ Zθ,µ}dµ=1, { Xθ,a }n a=1 } = TK(θ)K . The first property implies that ΩK(θ)(Zθ,µ, ·) = 0. The construction of Yθ,a yields (using (2.4), (2.5), (2.6), (3.7), (3.8), and (3.9)) ΩK(θ) ( Xθ,a, Yθ,b ) = 〈 Xθ,a, JK(θ)Yθ,b 〉 Rd+2n = X>θ,aJK(θ)Yθ,b = ( X>θ JK(θ)Yθ ) ab EJDE-2020/126 KAM TORI FOR PRESYMPLECTIC VECTOR FIELDS 11 = ( [ [X>θ X̃ > θ ] [ 0 X̃θRθ ]) ab = ( X̃>θ X̃θRθ ) ab = (In)ab = δab (In = (δab) is the unit n × n matrix), hence the vectors Xθ,a and Yθ,b form a symplectic basis of TK(θ)Q ∼= (TK(θ)P)/ ker ΩK(θ). We write this symbolically as ΩK(θ) (Xθ, Yθ) = X>θ JK(θ)Yθ = X̃>θ J̃K(θ)Ỹθ = In . (3.10) Below we summarize the properties of the basis of TK(θ)P constructed above, using matrix notations as in (3.10): Z>θ JK(θ)Zθ = Z̃>θ J̃K(θ)Z̃θ = 0 , Z>θ JK(θ)Xθ = Z̃>θ J̃K(θ)X̃θ = 0 , Z>θ JK(θ)Yθ = Z̃>θ J̃K(θ)Ỹθ = Z̃>θ X̃θRθ , X>θ JK(θ)Xθ = X̃>θ J̃K(θ)X̃θ = 0 , X>θ JK(θ)Yθ = X̃>θ J̃K(θ)Ỹθ = In , Y >θ JK(θ)Yθ = Ỹ >θ J̃K(θ)Ỹθ = −RθX̃>θ J̃−1 K(θ)X̃θRθ . (3.11) 3.2.3. Presymplecticity of V at K in adapted coordinates. If U, V,W ∈ X(P) with V presymplectic, i.e., LV Ω = 0 (recall Lemma 2.3), then 0 = (LV Ω)(U,W ) = LV ( Ω(U,W ) ) − Ω (LV U,W )− Ω (U,LVW ) = d+2n∑ A,B,C=1 UA (∂ΩAB ∂xC V C + ∂V C ∂xA ΩCB + ΩAC ∂V C ∂xB ) WB = d+2n∑ A,B=1 UA ( (DJ)V + (DV )>J + JDV )A BW B , where we used the operator J = ( JAB ) (2.4), lowering an index of a vector signifies transposition (i.e., contraction with the Euclidean metric tensor), and (DV )CB = ∂V C ∂xB , ( (DJ)V )A B := d+2n∑ C=1 ∂JAB ∂xC V C . Therefore in matrix notation the presymplecticity condition reads (DJ)V + (DV )>J + JDV = 0 . (3.12) Writing the derivative of V at K(θ) ∈ K as DVK(θ) =: ∂V∂x ∂V ∂x̃ ∂Ṽ ∂x ∂Ṽ ∂x̃  K(θ) , (3.13) we can easily show that (3.12) is equivalent to the conditions [∂Ṽ ∂x ] K(θ) = 0 , (DJ̃)K(θ)VK(θ) + [∂Ṽ ∂x̃ ]> K(θ) J̃K(θ) + J̃K(θ) [∂Ṽ ∂x̃ ] K(θ) = 0 . (3.14) 12 S. BAUER, N. P. PETROV EJDE-2020/126 3.3. Change of basis matrix Mθ. 3.3.1. Definition of Mθ. The adapted basis {Zθ,µ}dµ=1, {Xθ,a}na=1, {Yθ,a}na=1 of TK(θ)P constructed in Section 3.2 has properties that are very useful for our anal- ysis. Given an arbitrary column vector Uθ, considered as an element of TK(θ)P, we can find its components in the adapted basis as follows. Define the change of basis matrix Mθ of all vectors from the adapted basis, written as column vectors: Mθ := [ DKθ Yθ ] = [ Zθ Xθ Yθ ] = [ Zθ Xθ 0 Z̃θ X̃θ Ỹθ ] ∈ Md+2n,d+2n(R) . (3.15) Then the vector Uθ can be written as a superposition of the vectors from the adapted basis as follows: Uθ = Mθ ξθ = d∑ µ=1 Zθ,µ ξ µ θ + n∑ a=1 Xθ,a ξ d+a θ + n∑ a=1 Yθ,a ξ d+n+a θ . (3.16) In the adapted basis, if we write the (d+ 2n) components of the vector ξθ as three blocks of length d, n, and n, as in (3.16), then the vectors from TK(θ)K have the form ξθ = [∗ ∗ 0]>, where the stars represent numbers that are generally non-zero. 3.3.2. Computing (DVK(θ) − ∂ω)Mθ. In this section we will perform some compu- tations related to the change of basis matrix Mθ (3.15), which will be needed in Section 4. Differentiating the invariance condition (2.17), we obtain DVK(θ)DKθ = ∂ωDKθ . (3.17) This, together with the definition (3.15) of Mθ, gives us (DVK(θ) − ∂ω)Mθ = (DVK(θ) − ∂ω) [ DKθ Yθ ] = [ 0 0 (DVK(θ) − ∂ω)Yθ ] . Our first goal is to find an explicit expression for (DVK(θ) − ∂ω)Yθ. To this end we have to compute (DVK(θ) − ∂ω)Yθ = [ [∂V∂x̃ ]K(θ)Ỹθ [∂Ṽ∂x̃ ]K(θ)Ỹθ − ∂ωỸθ ] . By the Leibniz rule, ∂ωỸθ = ∂ω ( J̃−1 K(θ)X̃θRθ ) = ∂ω ( J̃−1 K(θ) ) X̃θRθ + J̃−1 K(θ)∂ω ( X̃θ ) Rθ + J̃−1 K(θ)X̃θ∂ωRθ . The elementary identity 0 = ∂ω ( I2n ) = ∂ω ( J̃−1 K(θ)J̃K(θ) ) , the invariance (2.17), and the presymplecticity condition (3.14) yield ∂ω ( J̃−1 K(θ) ) = −J̃−1 K(θ) ∂ω ( J̃K(θ) ) J̃−1 K(θ) = −J̃−1 K(θ)(DJ̃)K(θ)(∂ωKθ)J̃ −1 K(θ) = − ( J̃−1(DJ̃)V J̃−1 ) K(θ) = − ( J̃ [∂Ṽ ∂x̃ ] + [∂Ṽ ∂x̃ ]> J̃ ) K(θ) . (3.18) The invariance (3.17) and the expressions (3.13) and (3.14) give us ∂ωX̃θ = [∂Ṽ ∂x̃ ] K(θ) X̃θ . (3.19) From the definition (3.7) of Rθ, the expression (3.19) for ∂ωX̃θ, we easily obtain ∂ωRθ = −2RθX̃ > θ [∂Ṽ ∂x̃ ]sym K(θ) X̃θRθ , (3.20) EJDE-2020/126 KAM TORI FOR PRESYMPLECTIC VECTOR FIELDS 13 where [∂Ṽ ∂x̃ ]sym K(θ) := 1 2 ([∂Ṽ ∂x̃ ] K(θ) + [∂Ṽ ∂x̃ ]> K(θ) ) . It will be convenient to introduce the operator Π̃θ := I2n − X̃θRθX̃ > θ : R2n → R2n . (3.21) We collect some properties of Π̃θ and J̃−1 K(θ)Π̃θJ̃K(θ) = I2n − ỸθX̃ > θ J̃K(θ) which follow easily from (3.7), (3.8), and (3.11): • Π̃θ is symmetric; • both Π̃θ and J̃−1 K(θ)Π̃θJ̃K(θ) are idempotent; • the n columns of the matrix X̃θ are in the kernel of Π̃θ; • the n columns of X̃θ are eigenvectors of J̃−1 K(θ)Π̃θJ̃K(θ) with eigenvalue 1, while the n columns of Ỹθ are in the kernel of J̃−1 K(θ)Π̃θJ̃K(θ): J̃−1 K(θ)Π̃θJ̃K(θ) X̃θ = X̃θ , J̃−1 K(θ)Π̃θJ̃K(θ) Ỹθ = 0 , (3.22) hence J̃−1 K(θ)Π̃θJ̃K(θ) and ( I2n− J̃−1 K(θ)Π̃θJ̃K(θ) ) are projection operators cor- responding to the splitting TK(θ)Q = span { X̃θ,a }n a=1 ⊕ span { Ỹθ,a }n a=1 . Putting together (3.18), (3.19), and (3.20), and using the definition (3.21), we obtain ∂ωỸθ = [∂Ṽ ∂x̃ ] K(θ) Ỹθ + 2J̃−1 K(θ)Π̃θ [∂Ṽ ∂x̃ ]sym K(θ) X̃θRθ (3.23) so, finally, ( DVK(θ) − ∂ω ) Mθ = [ 0 0 [∂V∂x̃ ]K(θ)Ỹθ 0 0 −2J̃−1 K(θ)Π̃θ[ ∂Ṽ ∂x̃ ]sym K(θ)X̃θRθ ] . (3.24) 3.3.3. Writing (DVK(θ) − ∂ω)Mθ as MθCθ. Having computed (DVK(θ) − ∂ω)Mθ, we will rewrite it in a form that plays a crucial role in Section 4: (DVK(θ) − ∂ω)Mθ = MθCθ := Mθ 0 0 Tθ 0 0 Sθ 0 0 Uθ  . (3.25) Substitute (3.24) and (3.15) in (3.25) to obtain that Tθ, Sθ, and Uθ should satisfy ZθTθ +XθSθ = [∂V ∂x̃ ] K(θ) J̃−1 K(θ)X̃θRθ , (3.26) Z̃θTθ + X̃θSθ + ỸθUθ = −2J̃−1 K(θ)Π̃θ [∂Ṽ ∂x̃ ]sym K(θ) X̃θRθ . (3.27) Multiplying (3.27) separately by X̃>θ J̃K(θ) and Ỹ >θ J̃K(θ) on the left and using (3.11) and the definition of Rθ (3.7), we obtain Uθ = −2X̃>θ Π̃θ [∂Ṽ ∂x̃ ]sym K(θ) X̃θRθ , (3.28) −RθX̃>θ Z̃θTθ − Sθ −RθX̃>θ J̃−1 K(θ)X̃θRθUθ = −2Ỹ >θ Π̃θ [∂Ṽ ∂x̃ ]sym K(θ) X̃θRθ . (3.29) 14 S. BAUER, N. P. PETROV EJDE-2020/126 Since X̃>θ Π̃θ = (Π̃θX̃θ) > = 0, (3.28) yields Uθ = 0. From (3.26) and (3.29) we obtain Tθ = Z −1 θ ([∂V ∂x̃ ] K(θ) Ỹθ − 2XθỸ > θ Π̃θ [∂Ṽ ∂x̃ ]sym K(θ) X̃θRθ ) , Sθ = −Ỹ >θ J̃K(θ)Z̃θZ −1 θ [∂V ∂x̃ ] K(θ) Ỹθ + 2 ( In − Ỹ >θ J̃K(θ)Z̃θZ −1 θ Xθ ) Ỹ >θ Π̃θ [∂Ṽ ∂x̃ ]sym K(θ) X̃θRθ , (3.30) where we have set Z θ := Zθ −XθRθX̃ > θ Z̃θ = Zθ +XθỸ > θ J̃K(θ)Z̃θ ∈ Md,d(R) . (3.31) The geometric meaning of Z θ is discussed in detail in [9, Section 3.5.4]. We sum- marize our findings in the following lemma. Lemma 3.4. Let (P,Ω) be an exact presymplectic manifold, V ∈ X(P) be a presym- plectic vector field, K : Td+n → P be an invariant torus in the sense of Defini- tion 3.1, and the matrix Mθ be defined by (3.15). Then the equality (3.25) holds, with Uθ = 0 and Tθ and Sθ given by (3.30). 3.3.4. Factorization of Mθ. Later we will need the representation of M−1 θ that follows from the lemma below. Lemma 3.5. If the (d+ 2n)× (d+ 2n) matrices Qθ and Wθ are defined by Qθ := Id 0 0 X̃>θ 0 Ỹ >θ (Id 0 0 J̃K(θ) ) = Id 0 0 X̃>θ J̃K(θ) 0 Ỹ >θ J̃K(θ)  , (3.32) Wθ =  Zθ Xθ 0 0 0 In Ỹ >θ J̃K(θ)Z̃θ −In Ỹ >θ J̃K(θ)Ỹθ  , (3.33) then the following identity holds QθMθ = Wθ . (3.34) This implies that Mθ (3.15) is invertible if and only if Wθ is invertible. Proof. The columns of X̃θ and Ỹθ form a (symplectic) basis of R2n, which implies that the rows of X̃>θ and Ỹ >θ form a basis of R2n. Since J̃K(θ) is an invertible matrix (it corresponds to the symplectic form Ω̃ on Q, recall (2.3) and (2.5)), the rows of X̃>θ J̃K(θ) and Ỹ >θ J̃K(θ) from a basis of R2n, so that the matrix Qθ given by (3.32) is invertible. The identity (3.34) follows directly from (3.11). � 4. Approximate solutions In this section we examine the case when K is merely an approximate solution as defined below. We will build off of the results in Section 3 for true solutions to derive similar results for approximate solutions. We start with the definition for approximate solution. EJDE-2020/126 KAM TORI FOR PRESYMPLECTIC VECTOR FIELDS 15 Definition 4.1. Let (P,Ω) be an exact presymplectic manifold, {Vλ} be a (d+2n)- parameter family of presymplectic vector fields, ω ∈ D(γ, σ), and K0 : Td+n → P be an embedding. For a value λ0 of the parameter λ, define the error, e0,θ := Vλ0, K0(θ) − ∂ωK0,θ ∈ TK0(θ)P ∼= Rd+2n . (4.1) If some appropriately defined norm of e0 is sufficiently small, then we say that K0 is an approximate solution. 4.1. Approximately isotropic tori. In Lemma 3.3 we showed that if Kθ is a true solution (i.e., if (2.17) is satisfied), then the invariant manifold K = K(Td+n) is isotropic, i.e., K∗Ω = 0. The analogous result for this section will be that if K0,θ is an approximate solution, then K0 is approximately isotropic, i.e., K∗0 Ω is small. Lemma 4.2. Let (P,Ω) be an exact presymplectic manifold, {Vλ} be a (d + 2n)- parameter family of analytic presymplectic vector fields, and K0 ∈ Wρ (2.10) be an approximate solution with Diophantine frequency ω ∈ D(γ, σ). Assume that Vλ extends holomorphically to some complex neighborhood Br (2.18) of the image of Td+n ρ under K0, for some r > 0. Let L0,θ : TθTd+n → TθTd+n be the matrix representation of the pull-back (K∗0 Ω)θ as in (3.1) and (3.2): L0,θ = DK>0,θJK0(θ)DK0,θ . (4.2) Then there exists a constant C > 0 depending on d, n, σ, ρ, ‖DK0‖ρ, |Vλ0 |C1,Br , and |J |C1,Br , such that for every δ satisfying 0 < δ < ρ 2 , the following bound holds: ‖L0‖ρ−2δ < Cγ−1δ−(σ+1)‖e0‖ρ . (4.3) Proof. For the directional derivative of L0,θ (4.2), using (4.6), (4.1), and (3.12), we have ∂ωL0,θ = ∂ω ( DK>0,θJK0(θ)DK0,θ ) = ∂ω ( DK>0,θ ) JK0(θ)DK0,θ +DK>0,θ ∂ω ( JK0(θ) ) DK0,θ +DK>0,θ JK0(θ) ∂ω ( DK0,θ ) = ( DVλ0, K0(θ)DK0,θ −De0,θ )> JK0(θ)DK0,θ +DK>0,θDJK0(θ) ( Vλ0,K0(θ) − e0,θ ) DK0,θ +DK>0,θJK0(θ) ( DVλ0,K0(θ)DK0,θ −De0,θ ) = DK>0,θ ( DV >λ0, K0(θ)JK0(θ) +DJK0(θ) Vλ0, K0(θ) + JK0(θ)DVλ0, K0(θ) ) DK0,θ − ( De>0,θJK0(θ)DK0,θ +DK>0,θDJK0(θ)e0,θDK0,θ +DK>0,θJK0(θ)De0,θ ) = − ( De>0,θJK0(θ)DK0,θ +DK>0,θDJK0(θ)e0,θDK0,θ +DK>0,θJK0(θ)De0,θ ) . From this and the Cauchy bound (2.12) we obtain ‖∂ωL0‖ρ−δ ≤ C1‖e0‖ρ−δ + C2‖De0‖ρ−δ ≤ Cδ−1‖e0‖ρ . (4.4) Although K0 is only an approximate solution, the exactness of Ω implies that avg (L0) = 0 (the proof of this repeats part of the proof of Lemma 3.3, with K replaced by K0). We apply (4.4) and the Rüssmann estimate (2.13) to obtain ‖L0‖ρ−2δ ≤ Cγ−1δ−σ‖∂ωL0‖ρ−δ ≤ Cγ−1δ−(σ+1)‖e0‖ρ . � 16 S. BAUER, N. P. PETROV EJDE-2020/126 4.2. Linearized equation for the corrections. Given a family of presymplectic vector fields {Vλ}, the equation (2.17) can be difficult to solve for a value λ̄ of the parameter and an embedding K : Td+n → P. So instead of solving it directly for λ and K, we start with an approximate solution (λ0,K0) and construct an iterative process that produces better approximate solutions. As a result, we obtain a sequence (λj ,Kj) (1.2) that converges to (λ∞,K∞) = (λ̄,K). Let (λj ,Kj) be an approximate pair. Define the error (cf. (4.1)) ej,θ := Vλj ,Kj(θ) − ∂ωKj,θ ∈ TKj(θ)P ∼= Rd+2n . (4.5) We will usually consider ej as a mapping ej : Td+n → Rd+2n, whose derivative, Dej : Td+n → Md+2n,d+n(R), is given by Dej,θ = ( DVλj ,Kj(θ) − ∂ω ) DKj,θ ∈ Md+2n,d+n(R) . (4.6) Note that in (4.6), DVλj ,Kj(θ) stands for the derivative of the vector field Vλj with respect to the spatial variables x ∈ Rd+2n: DVλj ,Kj(θ) = [( DVλj ,Kj(θ) )A B ] = [∂V Aλj ,x ∂xB ∣∣ x=Kj(θ) ] ∈ Md+2n,d+2n(R) . Recall that the presymplecticity of Vλj imply that DVλj ,Kj(θ) (3.13) satisfies (3.14). Let εj and ∆j be (j + 1)st correction terms, i.e., λj+1 := λj + εj , Kj+1,θ := Kj,θ + ∆j,θ . (4.7) To derive an equation for the corrections εj and ∆j,θ, we set F [λ,K] := Vλ ◦K − ∂ωK, so that ej,θ = F [λj ,Kj ](θ). Expanding F [λj+1,Kj+1] about (λj ,Kj), we obtain ej+1,θ = ej,θ +DVλj ,Kj(θ) ∆j,θ − ∂ω∆j,θ + [∂Vλ ∂λ ] λj ,Kj(θ) εj +O(|∆j , εj |2) . (4.8) Therefore, if the corrections εj and ∆j satisfy the linear equation( DVλj , Kj(θ) − ∂ω ) ∆j,θ = −ej,θ − [∂Vλ ∂λ ] λj ,Kj(θ) εj , (4.9) the cancelations in the right-hand side of (4.8) guarantee quadratic convergence. The system (4.9) of (d+ 2n) equations for the corrections εj and ∆j is a linear algebraic equation with respect to εj , and a linear first-order partial differential equation with respect to ∆j . SinceDVλj ,Kj(θ) ∈ Md+2n,d+2n(R) is of a general form, is not easy to solve (4.9) and to obtain estimates on its solution. In Section 4.3 we will use the matrix Mθ (3.15) of change of basis from a general one to the adapted basis constructed in Section 3.2; the calculations from Section 3.3 will be very useful. 4.3. Solving the linearized equation. 4.3.1. Change of basis. We use an adapted basis in Rd+2n, so that instead of the unknown function ∆j we introduce the unknown function ξj : Td+n → Rd+2n through the linear change of basis ∆j,θ =: Mj,θ ξj,θ . (4.10) The change of basis matrix Mj,θ ∈ Md+2n,d+2n(R) is constructed similarly to the matrix Mθ in (3.15), but by using the approximate value λj and the approximate EJDE-2020/126 KAM TORI FOR PRESYMPLECTIC VECTOR FIELDS 17 embedding Kj . Namely, given an approximate invariant torus Kj (which we treat as a map Kj : Td+n → Rd+2n), we define[ Zj,θ Xj,θ Z̃j,θ X̃j,θ ] := Zj,θ Xj,θ] := DKj,θ ∈ Md+2n,d+n(R) as in (3.6). If the matrix X̃>j,θX̃j,θ is invertible (cf. Definition 4.3 below), define Rj,θ := ( X̃>j,θX̃j,θ )−1 ∈ Mn,n(R) as in (3.7), Ỹj,θ := J̃−1 Kj(θ) X̃j,θ Rj,θ ∈ M2n,n(R) , Yj,θ := [ 0 Ỹj,θ ] ∈ Md+2n,n(R) as in (3.8) and (3.9), and the approximate change of basis matrix Mj,θ as in (3.15): Mj,θ := [DKj,θ Yj,θ] = [Zj,θ Xj,θ Yj,θ] = ( Zj,θ Xj,θ 0 Z̃j,θ X̃j,θ Ỹj,θ ) ∈ Md+2n,d+2n(R) . Below we will need to invert the matrices Mj,θ for solving the linearized equation, which motivates the following definition. Definition 4.3. The map K ∈ Wρ is said to be a non-degenerate torus if the matrices X̃>θ X̃θ ∈ Mn,n(R) and Mθ ∈ Md+2n,d+2n(R) defined above (and therefore, the matrix Wθ as in (3.33)) are invertible. We will always assume that Kj is a non-degenerate torus (this is a part of Definition 4.7 below which is one of the assumptions in the Main Theorem). As before, we think of the columns Zj,θ,µ, Xj,θ,a, and Yj,θ,a (µ = 1, . . . , d, a = 1, . . . , n) of the matrices Zj,θ, Xj,θ, and Yj,θ as vectors in TKj(θ)P. If Kj is close to the true solution K, then these vectors still form a basis of TKj(θ)P as in the true case. By construction, the columns of Zj,θ and Xj,θ span the tangent space to the approximately invariant torus Kj := Kj(Td+n): span { {Zj,θ,µ}dµ=1, {Xj,θ,a}na=1 } = TKj(θ)Kj . (4.11) However, unlike the case of a true solution, Kj is not invariant with respect to the flow of Vλj , and ker ΩKj(θ) is generally not a subspace of TKj(θ)Kj . Making the substitution (4.10) in (4.9) and assuming that Mj,θ is invertible, we obtain the following equation for the new unknown function ξj,θ: M−1 j,θ ( DVλj ,Kj(θ)Mj,θ − ∂ωMj,θ ) ξj,θ − ∂ωξj,θ = −M−1 j,θ ( ej,θ + [∂Vλ ∂λ ] λj ,Kj(θ) εj ) . (4.12) To rewrite the coefficient of ξj,θ in (4.12) in a simple form, we want that (DVλj ,Kj(θ) − ∂ω)Mj,θ = Mj,θ (Cj,θ +Bj,θ) := Mj,θ (0 0 Tj,θ 0 0 Sj,θ 0 0 0 +Bj,θ ) (4.13) where Bj,θ is “small,” i.e., vanishing if ej becomes identically zero (cf. (3.25)). For the left-hand side, long and unenlightening calculations (cf. Section 3.3.2) yield( DVλj ,Kj(θ) − ∂ω ) Mj,θ = [ Dej,θ ( DVλj ,Kj(θ) − ∂ω ) Yj,θ ] 18 S. BAUER, N. P. PETROV EJDE-2020/126 = [( DVλj ,Kj(θ) − ∂ω ) DKj,θ ( DVλj ,Kj(θ) − ∂ω ) [ 0 Ỹj,θ ] ] = 0 0 [ ∂V λj ∂x̃ ]Kj(θ)[ ∂V λj ∂x̃ ]Kj(θ)Ỹj,θ 0 0 −2J̃−1 Kj(θ) Π̃j,θ[ ∂Ṽλj ∂x̃ ]sym Kj(θ) X̃j,θRj,θ + [ Dej,θ [ 0 E [ej ](θ) ] ] . Here we have set Π̃j,θ := I2n − X̃j,θRj,θX̃ > j,θ : R2n → R2n (cf. (3.21)), E [ej ](θ) := J̃−1 Kj(θ) DJ̃Kj(θ)ej,θỸj,θ − J̃ −1 Kj(θ) [∂ẽj,θ ∂θ ] Rj,θ + Ỹj,θ ([∂ẽj,θ ∂θ ]> X̃j,θ + X̃>j,θ [∂ẽj,θ ∂θ ]) Rj,θ ∈ M2n,n(R) , (4.14) where [ ∂ẽj,θ ∂θ ] is the 2n× n matrix with entries [∂ẽj,θ ∂θ ]i a := ∂ed+i j,θ ∂θd+a , i = 1, . . . , 2n; a = 1, . . . , n; and DJ̃Kj(θ)ej,θ stands for the 2n× 2n matrix with entries ( DJ̃Kj(θ)ej,θ )i k := d+2n∑ A=1 ∂J̃ ik ∂xA ∣∣ Kj(θ) eAj,θ , i, k = 1, . . . , 2n , For the entries of the matrix Cj,θ (4.13), we obtain (similarly to Section 3.3.3) Tj,θ = Z −1 j,θ ([∂V λj ∂x̃ ] Kj(θ) Ỹj,θ − 2Xj,θỸ > j,θΠ̃j,θ [∂Ṽλj ∂x̃ ]sym Kj(θ) X̃j,θRj,θ ) , Sj,θ = −Ỹ >j,θJ̃Kj(θ)Z̃j,θZ −1 j,θ [∂V λj ∂x̃ ] Kj(θ) Ỹj,θ + 2 ( In − Ỹ >j,θJ̃Kj(θ)Z̃j,θZ −1 j,θXj,θ ) Ỹ >j,θΠ̃j,θ [∂Ṽλj ∂x̃ ]sym Kj(θ) X̃j,θRj,θ , (4.15) where we have set Z j,θ := Zj,θ +Xj,θỸ > j,θJ̃Kj(θ)Z̃j,θ ∈ Md,d(R) (cf. (3.30), (3.31)). Summarizing, with the help of (4.13), (4.14), and (4.15), we rewrote (4.12) as (Cj,θ +Bj,θ) ξj,θ − ∂ωξj,θ = −M−1 j,θ ( ej,θ + [∂Vλ ∂λ ] λj ,Kj(θ) εj ) , (4.16) where Cj,θ is defined in (4.13) and (4.15), and Bj,θ is a “small” matrix, given by Bj,θ = M−1 j,θ [ Dej,θ [ 0 E [ej ](θ) ] ] . (4.17) 4.3.2. Invertibility issues. We define the (d+ 2n)× (d+ 2n) matrices Qj,θ and Wj,θ in exactly the same way as Qθ (3.32) and Wθ (3.33) but with Xθ, Yθ, Zθ, and JK(θ) replaced by Xj,θ, Yj,θ, Zj,θ, and JKj(θ), respectively. Since the rank of the (2n×2n) matrix [X̃j,θ Ỹj,θ] is maximal and J̃Kj(θ) is non-degenerate, Qj,θ is non-degenerate. By a direct calculation we obtain (cf. (3.34)) Qj,θMj,θ = Wj,θ + Pj,θ := Wj,θ +  0 0 0 X̃>j,θJ̃Kj(θ)Z̃j,θ X̃>j,θJ̃Kj(θ)X̃j,θ 0 0 0 0  , (4.18) where the matrix Pj,θ is small (if Kj were a true solution, Pj,θ would be zero). EJDE-2020/126 KAM TORI FOR PRESYMPLECTIC VECTOR FIELDS 19 Lemma 4.4. Assume that the hypotheses of Lemma 4.2 hold. Then there exists a constant C depending on d, n, σ, ρ, ‖DKj‖ρ, |Vλj |C1,Br , and |J |C1,Br , such that for every δ satisfying 0 < δ < ρ/2, we have the bound ‖W−1 j Pj‖ρ−2δ ≤ Cγ−1δ−(σ+1)‖ej‖ρ . (4.19) Proof. Recalling the bound (4.3) on the norm of the pull-back Lj (4.2) of the presymplectic form Ω to the torus Kj = Kj(Td+n), we obtain ‖W−1 j Pj‖ρ−2δ ≤ C‖Pj‖ρ−2δ ≤ C1‖X̃>j (J̃ ◦Kj)Z̃j‖ρ−2δ + C2‖X̃>j (J̃ ◦Kj)X̃j‖ρ−2δ ≤ C‖DK>j (J̃ ◦Kj)DKj‖ρ−2δ = C‖Lj‖ρ−2δ ≤ Cγ−1δ−(σ+1)‖ej‖ρ . � The approximate factorization (4.18) can be used to write the inverse matrix M−1 j,θ in a convenient form, and Lemma 4.4 yields some useful bounds. Lemma 4.5. Assume that the hypotheses of Lemma 4.2 hold. Let 0 < δ < ρ 2 and the error ej satisfy the bound Cγ−1δ−(σ+1)‖ej‖ρ ≤ 1 2 , (4.20) where C is the same constant as in (4.19). Then the matrix Mj,θ is invertible, and M−1 j,θ = W−1 j,θ Qj,θ +ME j,θ , (4.21) where the error term is ME j,θ = − ( Id+2n +W−1 j,θ Pj,θ )−1 W−1 j,θ Pj,θW −1 j,θ Qj,θ , (4.22) and satisfies the bound ‖ME j ‖ρ−2δ ≤ C ′γ−1δ−(σ+1)‖ej‖ρ ; (4.23) here C ′ is a constant that depends on the same parameters as the constant C in (4.19). Proof. The expression (4.22) comes from (4.18), and (4.23) follows from (4.19). � 4.3.3. Bounds on the “small” parts. Recall that, to find an approximate solution of the linearized equation (4.9), we changed the variable ∆j,θ to ξj,θ by (4.10) to trans- form it to the form (4.12). Then we rewrote the coefficient of ξj,θ in (4.12) as a sum of a “big” part, Cj,θ (given by (4.13) and (4.15)), and a “small” part, Bj,θ (4.17). In the Lemma below we give bounds on the “small” terms in (4.12). Lemma 4.6. Let Kj ∈ Wρ and the error ej be defined by (4.5). Let the pair (λj ,Kj) be non-degenerate (in the sense of Definition 4.7 below) for the family {Vλ} of presymplectic analytic vector fields. If the error ej satisfies (4.20), then the change of variables (4.10) transforms the linearized equation (4.9) to (Cj,θ +Bj,θ) ξj,θ − ∂ωξj,θ = −M−1 j,θ ( ej,θ + ∂λVλj ,Kj(θ) εj ) = −W−1 j,θ Qj,θej,θ −W −1 j,θ Qj,θ [∂Vλ ∂λ ] λj ,Kj(θ) εj −ME j,θej,θ −ME j,θ [∂Vλ ∂λ ] λj ,Kj(θ) εj , (4.24) 20 S. BAUER, N. P. PETROV EJDE-2020/126 where Cj,θ is defined by (4.13) and (4.15), Bj,θ by (4.17), ME j by (4.22). Further- more, ‖Bj‖ρ−2δ ≤ Cδ−1‖ej‖ρ , ‖ME j ej‖ρ−2δ ≤ Cγ−1δ−(σ+1)‖ej‖2ρ , ‖ME j [∂Vλ ∂λ ∣∣ λj ◦Kj ] εj‖ρ−2δ ≤ Cγ−1δ−(σ+1)‖∂Vλ ∂λ ∣∣ λj ◦Kj‖ ‖ej‖ρ|εj | . (4.25) Proof. Equation (4.24) follows directly from (4.16) and (4.21), so we only need to derive the bounds (4.25). Combining (4.17) and (4.21), we obtain Bj,θ = ( W−1 j,θ Qj,θ +ME j,θ ) [ Dej,θ ( 0 E [ej ](θ) )] . From the definition (4.14) of E [ej ](θ) and the Cauchy bound (2.12), ‖E [ej ]‖ρ−2δ ≤ C1‖ej‖ρ−2δ + C2δ −1‖ej‖ρ−δ ≤ Cδ−1‖ej‖ρ−δ, which, together with the bound (4.23) on ME j , yields the first bound in (4.25): ‖Bj‖ρ−2δ ≤ ( ‖W−1 j Qj‖ρ−2δ + ‖ME j ‖ρ−2δ ) (‖Dej‖ρ−2δ + ‖E [ej ]‖ρ−2δ) ≤ ( C1 + C2γ −1δ−(σ+1)‖ej‖ρ ) γ−1‖ej‖ρ−δ ≤ Cγ−1‖ej‖ρ . The remaining two bounds in (4.25) in are direct consequences of (4.23). � 4.3.4. Solving the simplified equation. To use the Newton method for finding ξj,θ, it is enough to solve (4.24) retaining only the “big” terms, i.e., ignoring all terms of higher order with respect to ‖ej‖. As we will show below (see (4.37)), the term εj is of order of ‖ej‖. Lemma 4.6 allows us to keep only the leading terms in (4.24): Cj,θξj,θ − ∂ωξj,θ = −W−1 j,θ Qj,θej,θ − Λj,θεj , (4.26) where we have set Λj,θ := W−1 j,θ Qj,θ [∂Vλ ∂λ ] λj ,Kj(θ) . (4.27) Let us denote the first d components of ξj,θ by ξz j,θ, the next n components by ξx j,θ, and the last n components by ξy j,θ. Using the specific form of Cj,θ (4.13), we rewrite (4.26) in the form ∂ω ξz j,θ ξx j,θ ξy j,θ  = W−1 j,θ Qj,θej,θ + Λj,θ εj + Tj,θ ξy j,θ Sj,θ ξ y j,θ 0  . (4.28) Integrating both sides of (4.28) over Td+n, we obtain that (4.28) has a solution if and only if the average over Td+n of its right-hand side is 0: avg ( W−1 j Qjej ) + avg (Λj) εj + avg(Tj ξ y j ) avg(Sj ξ y j ) 0  = 0 . (4.29) Observe that the right-hand side of the last n equations of the system (4.28) does not involve ξy j , so that the last n equations of (4.28) have the form ∂ωξ y j,θ = ( W−1 j,θ Qj,θej,θ + Λj,θ εj )y , (4.30) EJDE-2020/126 KAM TORI FOR PRESYMPLECTIC VECTOR FIELDS 21 where the superscript y stands for the last n components. The system (4.30) for ξy j,θ has a solution if and only if the last-hand side of (4.30) has zero average. To satisfy this condition, we will require initially that εj satisfy the linear system avg ( W−1 j Qjej ) + avg (Λj) εj = 0 , (4.31) which in turn has a solution if and only if the matrix avg (Λj) is invertible. In order to guarantee that we can solve (4.31) for εj , we introduce the following Definition 4.7. The pair (λ,K) is said to be non-degenerate for a (d + 2n)- parameter family of vector fields Vλ if K ∈ Wρ is a non-degenerate torus (recall Definition 4.3) and the matrix Λ (as in (4.27)) has a non-singular average: rank avg (Λ) = d+ 2n , avg (Λ) := ∫ Td+n Λθ dθ1 · · · dθd+n . (4.32) We assume that the pair (λj ,Kj) is non-degenerate, and set εj to be equal to the preliminary value εprelim j := −{avg (Λj)}−1 avg ( W−1 j Qjej ) , (4.33) so that (4.31) is satisfied; εprelim j satisfies the bound |εprelim j | ≤ C avg (ej) ≤ C‖ej‖ρ . (4.34) This choice of εj guarantees the existence of a solution ξy j of (4.30) which (thanks to (4.34) and the Rüssmann’s inequality (2.13)) satisfies the bound ‖ξy j ‖ρ−δ ≤ Cγ −1δ−σ ∥∥W−1 j Qjej + Λjε prelim j ∥∥ ρ ≤ Cγ−1δ−σ‖ej‖ρ . (4.35) Having found ξy j from solving (4.30), we redefine εj as εj := −{avg (Λj)}−1 ( avg ( W−1 j Qjej ) + avg(Tj ξ y j ) avg(Sj ξ y j ) 0 ) (4.36) to satisfy the solvability condition (4.29) for (4.28). Note that the change from εprelim j (4.33) to εj (4.36) does not affect the component ξy j . Thanks to (4.35), εj satisfies |εj | ≤ C ( ‖ej‖ρ−δ + ‖ξy j ‖ρ−δ ) ≤ Cγ−1δ−σ‖ej‖ρ . (4.37) With the new value of εj from (4.36), we solve (4.28) to find ξj which, according to the Rüssmann’s inequality (2.13) and the bound (4.37), satisfies ‖ξj‖ρ−2δ ≤ Cγ−1δ−σ ( ‖W−1 j Qjej + Λjεj‖ρ−δ + C‖ξy j ‖ρ−δ ) ≤ Cγ−1δ−σ ( ‖ej‖ρ + |εj |+ γ−1δ−σ‖ej‖ρ ) ≤ Cγ−2δ−2σ‖ej‖ρ . (4.38) We summarize our findings in the following lemma. Lemma 4.8. Assume the hypotheses of Lemma 4.6. Then there exist a parameter εj and a function ξj that solve the reduced linear equation (4.26) and satisfy the bounds (4.37) and (4.38). 22 S. BAUER, N. P. PETROV EJDE-2020/126 5. Newton’s method In this section we will collect the estimates for the jth step of the iterative scheme and show that the Newton method generates a Cauchy sequence of approximate solutions in a Banach space which converges to a true solution. We only give brief sketches, referring the reader to [21, 3] for details. Lemma 5.1. If the assumptions of Lemma 4.8 are satisfied and rj := ‖Kj − K0‖ρj < r, then there exist a function ∆j and a parameter εj ∈ Rd+2n such that ‖∆j‖ρj−2δj ≤ cjγ−2δ−2σ j ‖ej‖ρj ‖D∆j‖ρj−3δj ≤ cjγ−2δ −(2σ+1) j ‖ej‖ρj |εj | ≤ cj ∣∣avg(Λj) −1 ∣∣ ‖ej‖ρj , (5.1) where cj is a constant that depends on n, d, r, ρ, |Vλj |C2,Br , ‖DKj‖ρj , ‖Rj‖ρj , and ‖∂Vλ∂λ ‖ρj Additionally, if rj + cjγ −2δ −(2σ−1) j ‖ej‖ρj < r , (5.2) then ‖ej+1‖ρj+1 ≤ cjγ−4δ−4σ j ‖ej‖2ρj . (5.3) Proof. The inequalities in (5.1) follow from Lemmata 2.7 and 4.6, and (4.10). To see that Kj+1 ∈ Br, that is, Kj+1 stays within the neighborhood where V is holomorphically extended, we use (5.1) and (5.2): ‖Kj+1 −K0‖ρj+1−2δj+1 = ‖Kj + ∆j −K0‖ρj+1−2δj+1 ≤ ‖Kj −K0‖ρj + ‖∆j‖ρj−2δj ≤ rj + cjγ −2δ −(2σ+1) j ‖ej‖ρj < r . To prove (5.3), recall from (4.8) that ξj = M−1 j ∆j was found by solving (4.26), so DVλj ,Kj(θ) ∆j,θ − ∂ω∆j,θ + [∂Vλ ∂λ ] λj ,Kj(θ) εj + ej = Mj,θ ( Bj,θξj,θ +ME j,θej,θ +ME j,θ [∂Vλ ∂λ ∣∣ λj ◦Kj ]) , and each term on the right hand side is quadratically small from Lemma 4.6, hence∥∥DVλj ,Kj ∆j − ∂ω∆j + [∂Vλ ∂λ ] λj ,Kj εj + ej ∥∥ ρj−2δj ≤ Cγ−3δ−(3σ+1)‖ej‖2ρj . Finally, recalling the Taylor expansion (4.8) of ej+1,θ, we see that the remainder term is on the order of ‖∆j‖2ρj−2δj . Thus we get the estimate (5.3). � The lemma below guarantees that, if the error is small and some invertibility conditions are met at the jth step, then the invertibility holds at the (j+ 1)st step. Lemma 5.2. Assume the setup of Lemma 5.1. If cjγ −2δ −(σ+1) j ‖ej‖ρj ≤ 1/2, then: (i) if X̃>j X̃j is invertible, then X̃>j+1X̃j+1 is invertible; (ii) if Wj is invertible, then Wj+1 is invertible; (iii) if avg(Λj) is invertible, then avg(Λj+1) is invertible. EJDE-2020/126 KAM TORI FOR PRESYMPLECTIC VECTOR FIELDS 23 Proof. Recalling that X̃j is a part of the matrix DKj , we obtain X̃>j+1X̃j+1 = X̃>j X̃j +Pj , with Pj := X̃>j ∆̃j,x̃ + ∆̃>j,x̃X̃j + ∆̃>j,x̃∆̃j,x̃, where ∆̃j,x̃ ∈ M2n,2n(R) has entries (∆̃j,x̃)ik = ∂(∆̃j) i/∂x̃k. The bounds (5.1) give an upper bound on the size of Pj . The matrix X̃>j X̃j is invertible by assumption, In+ ( X̃>j X̃j )−1 Pj is invertible by the Neumann series, so X̃>j+1X̃j+1 = X̃>j X̃j ( In + ( X̃>j X̃j )−1 Pj ) is invertible, which completes the proof of (i). The proofs of (ii) and (iii) are similar. � The lemma below shows how close the initial approximation has to be for the Newton method to be iterated indefinitely and to converge to a true solution K∞, and gives a bound on the difference between K∞ and the initial approximation K0. The proof can be found in [21, Lemma 13]. Lemma 5.3. Let {cj}j≥0 be the sequence of constants from Lemmata 5.1 and 5.2. For 0 < δ0 < min(ρ0/12 , 1) we define δj := δ02−j , ρj := ρj−1 − 6δj−1 , rj := ‖Kj −K0‖ρj , ρ∞ := limj→∞ ρj, K∞ := limj→∞Kj. Then there exists a constant C > 0 depend- ing on d, n, |Vλ|C2,Br , |J0|C1,Br , ‖DK0‖ρ0 , and |{avg(Λ0)}−1| such that if ‖e0‖ρ0 satisfies the conditions C24σγ−4δ−4σ 0 ‖e0‖ρ0 ≤ 1 2 , C ( 1 + 24σ 22σ − 1 ) γ−2δ−2σ 0 ‖e0‖ρ0 < r , then the Newton method converges to a true solution (λ∞,K∞). Furthermore, ‖K∞ −K0‖ρ0−6δ0 ≤ 22σ 22σ − 1 cγ−2δ−2σ 0 ‖e0‖ρ0 . Acknowledgments. The research of the authors was partially supported by NSF grant DMS-0807658. NPP also acknowledges the generous support by the Nancy Scofield Hester Presidential Professorship. We would like to thank Rafael de la Llave, Garrett Alston, and Mahesh Sunkula for encouragement and enlightening discussions. References [1] R. Abrahamm J. E. Marsden; Foundations of Mechanics, second ed., Benjamin/Cummings, Reading, Mass., 1978. MR 515141 (81e:58025) [2] L. Abrunheiro, M. Camarinha, J. Clemente-Gallardo; Cubic polynomials on Lie groups: re- duction of the Hamiltonian system, J. Phys. A 44 (2011), no. 35, 355203, 16, Corrigendum: 46(18):189501, 2, 2013. MR 2823461 (2012k:37116) [3] H. N. Alishah, R. de la Llave; Tracing KAM tori in presymplectic dynamical systems, J. Dynam. Differential Equations 24 (2012), no. 4, 685–711. MR 3000600 [4] J. L. Anderson, P. G. Bergmann; Constraints in covariant field theories, Physical Rev. (2) 83 (1951), 1018–1025. MR 0044382 [5] V. I. Arnold, V. V. Kozlov, A. I. Neishtadt; Mathematical Aspects of Classical and Celestial Mechanics, third ed., Encyclopaedia of Mathematical Sciences, vol. 3, Springer-Verlag, Berlin, 2006, [Dynamical systems. III]. MR 2269239 [6] M. Barbero-Liñán, M. C. Muñoz-Lecanda; Geometric approach to Pontryagin’s maximum principle, Acta Appl. Math. 108 (2009), no. 2, 429–485. MR 2551483 [7] C. Batlle, J. Gomis, J. M. Pons, N. Roman; Lagrangian and Hamiltonian constraints, Lett. Math. Phys. 13 (1987), no. 1, 17–23. MR 878657 (88h:58041) 24 S. BAUER, N. P. PETROV EJDE-2020/126 [8] C. Batlle, J. Gomis, J. M. Pons, N. Román-Roy; Equivalence between the Lagrangian and Hamiltonian formalism for constrained systems, J. Math. Phys. 27 (1986), no. 12, 2953–2962. MR 866595 (88a:58060) [9] S. Bauer; On the existence of KAM Tori for Presymplectic Vector Fields, ProQuest LLC, Ann Arbor, MI, 2016, Thesis (Ph.D.)–The University of Oklahoma. [10] P. G. Bergmann, I. Goldberg; Dirac bracket transformations in phase space, Phys. Rev. (2) 98 (1955), 531–538. MR 0074314 [11] H. Broer, M. Sevryuk; KAM theory: quasi-periodicity in dynamical systems, Handbook of Dynamical Systems. Vol. 3, Elsevier/North-Holland, Amsterdam, 2010, pp. 249–344. MR 3292649 [12] R. Calleja, R. de la Llave; A numerically accessible criterion for the breakdown of quasi- periodic solutions and its rigorous justification, Nonlinearity 23 (2010), no. 9, 2029–2058. MR 2672635 [13] R. C. Calleja, A. Celletti, R. de la Llave; A KAM theory for conformally symplectic systems: efficient algorithms and their validation, J. Differential Equations 255 (2013), no. 5, 978– 1049. MR 3062760 [14] J. F. Cariñena, J. Gomis, L. A. Ibort, N. Román; Canonical transformations theory for presymplectic systems, J. Math. Phys. 26 (1985), no. 8, 1961–1969. MR 796226 (86j:58041) [15] J. F. Cariñena, L. A. Ibort; Geometric theory of the equivalence of Lagrangians for con- strained systems, J. Phys. A 18 (1985), no. 17, 3335–3341. MR 817862 [16] J. F. Cariñena, C. Lopez, N. Román-Roy; Geometric study of the connection between the La- grangian and Hamiltonian constraints, J. Geom. Phys. 4 (1987), no. 3, 315–334. MR 957017 (89i:58031) [17] J. F. Cariñena, J. Nasarre; On the symplectic structures arising in geometric optics, Fortschr. Phys. 44 (1996), no. 3, 181–198. MR 1400305 [18] J. F. Cariñena, J. Nasarre; Presymplectic geometry and Fermat’s principle for anisotropic media, J. Phys. A 29 (1996), no. 8, 1695–1702. MR 1395798 [19] L. Chierchia; KAM lectures, Dynamical Systems. Part I, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 2003, Hamiltonian systems and celestial mechanics, Selected papers from the Research Trimester held in Pisa, February 4–April 26, 2002, pp. 1–55. MR 2071231 [20] L. Chierchia; Kolmogorov-Arnold-Moser (KAM) theory, Encyclopedia of Complexity and Systems Science (R. A. Meyers, ed.), Springer, 2009, pp. 5064–5091. [21] R. de la Llave, A. González, À. Jorba, J. Villanueva; KAM theory without action-angle variables, Nonlinearity 18 (2005), no. 2, 855–895. MR 2122688 (2005k:37131) [22] M. de León, D. Mart́ın de Diego; Almost product structures and Poisson reduction of presym- plectic systems, Extracta Math. 10 (1995), no. 1, 37–45. MR 1359590 [23] P. A. M. Dirac; Generalized Hamiltonian dynamics, Canadian J. Math. 2 (1950), 129–148. MR 0043724 (13,306b) [24] P. A. M. Dirac; The Hamiltonian form of field dynamics, Canadian J. Math. 3 (1951), 1–23. MR 0043725 (13,306c) [25] P. A. M. Dirac; Generalized Hamiltonian dynamics, Proc. Roy. Soc. London. Ser. A 246 (1958), 326–332. MR 0094205 (20 #724) [26] H. S. Dumas; The KAM story: A Friendly Introduction to the Content, History, and Signif- icance of Classical Kolmogorov-Arnold-Moser Theory, World Scientific, New Jersey, 2014. [27] C. Duval; Polarized spinoptics and symplectic physics, Preprint (2013), arXiv:1312.4486 [math-ph]. [28] A. Echeverŕıa-Enŕıquez, J. Maŕın-Solano, M. C. Muñoz-Lecanda, N. Román-Roy; Geometric reduction in optimal control theory with symmetries, Rep. Math. Phys. 52 (2003), no. 1, 89–113. MR 2006728 (2004h:49037) [29] A. Echeverŕıa-Enŕıquez, M. C. Muñoz-Lecanda, N. Román-Roy; Reduction of presymplectic manifolds with symmetry, Rev. Math. Phys. 11 (1999), no. 10, 1209–1247. MR 1734712 (2001b:53106) [30] E. Fontich, R. de la Llave, Y. Sire; Construction of invariant whiskered tori by a parameteri- zation method. I. Maps and flows in finite dimensions, J. Differential Equations 246 (2009), no. 8, 3136–3213. MR 2507954 (2010c:37139) [31] A. M. Fox, J. D. Meiss; Critical invariant circles in asymmetric and multiharmonic gener- alized standard maps, Commun. Nonlinear Sci. Numer. Simul. 19 (2014), no. 4, 1004–1026. MR 3119277 EJDE-2020/126 KAM TORI FOR PRESYMPLECTIC VECTOR FIELDS 25 [32] A. González-Enŕıquez, A. Haro, R. de la Llave; Singularity theory for non-twist KAM tori, Mem. Amer. Math. Soc. 227 (2014), no. 1067, vi+115. MR 3154587 [33] M. Gotay; Presymplectic Manifolds, Geometric Constraint Theory and Dirac-Bergmann Theory of Constraints, Ph.D. thesis, University of Maryland, 1979. [34] M. J. Gotay, J. M. Nester; Presymplectic Lagrangian systems. I. The constraint algorithm and the equivalence theorem, Ann. Inst. H. Poincaré Sect. A (N.S.) 30 (1979), no. 2, 129–142. MR 535369 (80j:58035) [35] M. J. Gotay, J. M. Nester, G. Hinds; Presymplectic manifolds and the Dirac-Bergmann theory of constraints, J. Math. Phys. 19 (1978), no. 11, 2388–2399. MR 506712 (80e:58025) [36] A. Hanson, T. Regge, C. Teitelboim; Constrained Hamiltonian Systems, Accademia Nazionale dei Lincei, Roma, 1976. [37] À. Haro, M. Canadell, A. Luque, J. M. Mondelo, J.-L. Figueras; The Parameterization Method for Invariant Manifolds: From Rigorous Results to Effective Computations, Applied Mathematical Sciences, vol. 195, Springer-Verlag, 2016. [38] M. Henneaux, C. Teitelboim; Quantization of Gauge Systems, Princeton University Press, Princeton, NJ, 1992. MR 1191617 (94h:81003) [39] G. Huguet, R. de la Llave, Y. Sire; Computation of whiskered invariant tori and their associ- ated manifolds: new fast algorithms, Discrete Contin. Dyn. Syst. 32 (2012), no. 4, 1309–1353. MR 2851901 (2012k:37126) [40] P. Libermann, C.-M. Marle; Symplectic Geometry and Analytical Mechanics, D. Reidel, Dor- drecht, 1987. MR 882548 (88c:58016) [41] R. de la Llave; A tutorial on KAM theory, Smooth Ergodic Theory and Its Applications (Seattle, WA, 1999), Proc. Sympos. Pure Math., vol. 69, Amer. Math. Soc., Providence, RI, 2001, pp. 175–292. MR 2002h:37123 [42] A. Luque, J. Villanueva; A KAM theorem without action-angle variables for elliptic lower dimensional tori, Nonlinearity 24 (2011), no. 4, 1033–1080. MR 2773779 [43] J. Moser; Stable and Random Motions in Dynamical Systems, Annals of Mathematics Stud- ies, No. 77, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1973. MR 0442980 [44] E. Newman, P. G. Bergmann; Lagrangians linear in the “velocities”, Phys. Rev. (2) 99 (1955), 587–592. MR 0072039 [45] T. Noda, M. Oda; Geometry of moment maps and reductions for presymplectic manifolds, Georgian Math. J. 16 (2009), no. 2, 343–352. MR 2562366 [46] J. Pöschel; A lecture on the classical KAM theorem, Smooth Ergodic Theory and Its Applica- tions (Seattle, WA, 1999), Proc. Sympos. Pure Math., vol. 69, Amer. Math. Soc., Providence, RI, 2001, pp. 707–732. MR 1858551 [47] H. J. Rothe, K. D. Rothe; Classical and Quantum Dynamics of Constrained Hamiltonian Systems, World Scientific Lecture Notes in Physics, vol. 81, World Scientific, Hackensack, NJ, 2010. MR 2723931 [48] G. Rudolph, M. Schmidt; Differential Geometry and Mathematical Physics, I: Manifolds, Lie Grouips and Hamiltonian Systems, Springer, 2013. [49] H. Rüssmann; On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus, Dynamical Systems, Theory and Ap- plications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974), Lecture Notes in Phys., vol. 38, Springer, Berlin, 1975, pp. 598–624. MR 0467824 (57 #7675) [50] D. A. Salamon; The Kolmogorov-Arnold-Moser theorem, Math. Phys. Electron. J. 10 (2004), Paper 3, 37 pp. (electronic). MR 2111297 (2005h:37135) [51] M. B. Sevryuk; The classical KAM theory at the dawn of the twenty-first century, Mosc. Math. J. 3 (2003), no. 3, 1113–1144, 1201–1202. MR 2078576 [52] E. C. G. Sudarshan, N. Mukunda; Classical Dynamics: A Modern Perspective, Wiley- Interscience, New York-London-Sydney, 1974. MR 0434047 (55 #7016) [53] K. Sundermeyer; Constrained Dynamics, Lecture Notes in Physics, vol. 169, Springer-Verlag, Berlin-New York, 1982. MR 678773 (84f:58051) Sean Bauer Department of Mathematics, University of Oklahoma, Norman, OK 73019, USA. American Fidelity, 9000 Cameron Parkway, Oklahoma City, OK 73114, USA Email address: sean.michael.bauer@gmail.com 26 S. BAUER, N. P. PETROV EJDE-2020/126 Nikola P. Petrov Department of Mathematics, University of Oklahoma, Norman, OK 73019, USA Email address: npetrov@ou.edu 1. Introduction 2. Preliminaries and general setup 2.1. Presymplectic manifolds and vector fields 2.2. Foliation induced by ker 2.3. Matrix representation of and "0365 2.4. Miscellaneous definitions and results 2.5. General setup and matrix notation 2.6. Statement of the main theorem 3. Exact solutions 3.1. Invariant tori are isotropic 3.2. Construction and properties of an adapted basis of TK() P 3.3. Change of basis matrix M 4. Approximate solutions 4.1. Approximately isotropic tori 4.2. Linearized equation for the corrections 4.3. Solving the linearized equation 5. Newton's method Acknowledgments References