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Classification of Arnold-Beltrami Flows and their Hidden Symmetries P. Freá,c,1 and A.S. Sorinb,c aDipartimento di Fisica, Universitá di Torino INFN – Sezione di Torino via P. Giuria 1, 10125 Torino Italy e-mail: [email protected] bBogoliubov Laboratory of Theoretical Physics & Veksler and Baldin Laboratory of High Energy Physics 5 Joint Institute for Nuclear Research, 1 141980 Dubna, Moscow Region, Russia 0 e-mail: [email protected] 2 n cNational Research Nuclear University MEPhI a (Moscow Engineering Physics Institute), J Kashirskoye shosse 31, 115409 Moscow, Russia 9 1 ] h p - Abstract h Inthecontextofmathematicalhydrodynamics,weconsiderthegrouptheorystructurewhichunderlies t a thesonamedABCflowsintroducedbyBeltrami,ArnoldandChildress. MainreferencepointsareArnold’s m theorem stating that, for flows taking place on compact three manifolds M , the only velocity fields 3 [ able to produce chaotic streamlines are those satisfying Beltrami equation and the modern topological conception of contact structures, each of which admits a representative contact one-form also satisfying 1 v Beltramiequation.WeadvocatethatBeltramiequationisnothingelsebuttheeigenstateequationforthe 4 first order Laplace-Beltrami operator (cid:63) d, which can be solved by using time-honored harmonic analysis. g 0 TakingforM atorusT3 constructedasR3/Λ,whereΛisacrystallographiclattice,wepresentageneral 3 6 algorithm to construct solutions of the Beltrami equation which utilizes as main ingredient the orbits 4 under the action of the point group P of three-vectors in the momentum lattice (cid:63)Λ. Inspired by the 0 Λ . crystallographic construction of space groups, we introduce the new notion of a Universal Classifying 1 Group GU which contains all space groups as proper subgroups. We show that the (cid:63) d eigenfunctions 0 Λ g 5 are naturally arranged into irreducible representations of GUΛ and by means of a systematic use of the 1 branching rules with respect to various possible subgroups Hi ⊂GUΛ we search and find Beltrami fields : withnontrivialhiddensymmetries. Inthecaseofthecubiclatticethepointgroupistheproperoctahedral v i group O24 and the Universal Classifying Group GUcubic is a finite group G1536 of order |G1536| = 1536 X which we studyin fulldetail derivingall of its37 irreduciblerepresentationsand theassociated character r table. We show that the O orbits in the cubic lattice are arranged into 48 equivalence classes, the a 24 parameters of the corresponding Beltrami vector fields filling all the 37 irreducible representations of G . Inthiswayweobtainanexhaustiveclassificationofallgeneralized ABC-flows andoftheirhidden 1536 symmetries. We make several conceptual comments about the need of a field-theory yielding Beltrami equationasafieldequationand/oraninstantonequationandonthepossiblerelationofArnold-Beltrami flows with (supersymmetric) Chern-Simons gauge theories. We also suggest linear generalizations of Beltramiequationtohigherodd-dimensionsthataredifferentfromthenon-linearoneproposedbyArnold and possibly make contact with M-theory and the geometry of flux-compactifications. 1Prof. FréispresentlyfulfillingthedutiesofScientificCounseloroftheItalianEmbassyintheRussianFederation,Denezhnij pereulok, 5, 121002 Moscow, Russia. e-mail: [email protected] Contents I The Article 4 1 Introduction 4 1.1 Beltrami Flows and Arnold Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 The conception of contact structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Beltrami equation at large and harmonic analysis . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 Harmonic analysis on the T3 torus and the Universal Classifying Group . . . . . . . . . . . . 11 1.5 Organization of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2 Basic Elements of Lattice and Finite Group Theory needed in our construction 14 2.1 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 The three torus T3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Bravais Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4 The proper Point Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.5 Point Group Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3 The spectrum of the (cid:63)d operator on T3 and Beltrami equation 17 3.1 The algorithm to construct Arnold Beltrami Flows . . . . . . . . . . . . . . . . . . . . . . . . 19 4 The Cubic Lattice and the Octahedral Point Group 21 4.1 Structure of the Octahedral Group O ∼ S . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 24 4 4.2 Irreducible representations of the Octahedral Group . . . . . . . . . . . . . . . . . . . . . . . 23 4.2.1 D : the identity representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1 4.2.2 D : the quadratic Vandermonde representation . . . . . . . . . . . . . . . . . . . . . 24 2 4.2.3 D : the two-dimensional representation . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3 4.2.4 D : the three-dimensional defining representation . . . . . . . . . . . . . . . . . . . . 24 4 4.2.5 D : the three-dimensional unoriented representation . . . . . . . . . . . . . . . . . . . 25 5 5 Extension of the Point Group with Translations and more Group Theory 25 5.1 The idea of Space Groups and Frobenius congruences . . . . . . . . . . . . . . . . . . . . . . 26 5.1.1 Frobenius congruences for the Octahedral Group O . . . . . . . . . . . . . . . . . . . 27 24 5.2 The Universal Classifying Group for the cubic lattice: G . . . . . . . . . . . . . . . . . . . 29 1536 5.3 Structure of the G group and derivation of its irreps . . . . . . . . . . . . . . . . . . . . . 30 1536 5.3.1 Strategy to construct the irreducible representations of a solvable group . . . . . . . . 30 5.3.2 The inductive algorithm for irreps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.3.3 Derivation of G irreps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1536 6 The spherical layers and the octahedral lattice orbits 33 6.1 Classification of the 48 types of orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 7 Discussion of explicit examples of Arnold-Beltrami Flows from Octahedral Point Orbits 39 7.1 The lowest lying octahedral orbit of length 6 in the cubic lattice and the ABC-flows . . . . . 39 7.1.1 The (A,A,A)-flow invariant under GS . . . . . . . . . . . . . . . . . . . . . . . . . . 42 24 7.1.2 Chains of subgroups and the flows (A,B,0), (A,A,0) and (A,0,0) . . . . . . . . . . . 44 7.2 The lowest lying octahedral orbit of length 12 in the cubic lattice and the Beltrami flows respectively invariant under GP and GK . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 24 24 1 7.2.1 Beltrami Flows invariant with respect to the subgroups GP and GK . . . . . . . . 51 24 24 7.3 The lowest lying octahedral orbit of length 8 in the cubic lattice . . . . . . . . . . . . . . . . 54 7.3.1 Beltrami vector fields with hidden symmetry GS and GK , respectively . . . . . . . 57 32 32 7.4 Example of an octahedral orbit of length 24 in the cubic lattice . . . . . . . . . . . . . . . . . 60 7.4.1 Point Group irreps and Uplifting to the Universal Classifying Group . . . . . . . . . . 62 7.4.2 The Beltrami flow invariant under Oh . . . . . . . . . . . . . . . . . . . . . . . . . . 64 48 8 The Hexagonal Lattice and the Dihedral Group D 64 6 8.1 The dihedral group D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6 8.2 Irreducible representations of the dihedral group D and the character table . . . . . . . . . . 68 6 8.2.1 D : the identity representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 1 8.2.2 D : the second one-dimensional representation . . . . . . . . . . . . . . . . . . . . . . 68 2 8.2.3 D : the third one-dimensional representation . . . . . . . . . . . . . . . . . . . . . . . 68 3 8.2.4 D : the fourth one-dimensional representation . . . . . . . . . . . . . . . . . . . . . . 69 4 8.2.5 D : the first two-dimensional representation . . . . . . . . . . . . . . . . . . . . . . . . 69 5 8.2.6 D : the second two-dimensional representation . . . . . . . . . . . . . . . . . . . . . . 69 6 8.3 Spherical layers in the Hexagonal lattice and D orbits . . . . . . . . . . . . . . . . . . . . . . 70 6 9 Beltrami Fields from spherical layers in the Hexagonal Lattice 71 9.1 The lowest lying layer of length 8 in the hexagonal lattice Λ(cid:63) . . . . . . . . . . . . . . . . . 71 Hex 9.2 The lowest lying orbit of length 12 in the hexagonal lattice Λ(cid:63) . . . . . . . . . . . . . . . . 75 Hex 10 Conclusions and Outlook 80 II The Appendices: tables of conjugacy classes, characters, branching rules and orbit splittings 86 A Description of the Universal Classifying Group G and of its subgroups 86 1536 A.1 The Group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 1536 A.2 The Group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 768 A.3 The Group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 256 A.4 The Group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 128 A.5 The group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 64 A.6 The Group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 192 A.7 The Group GF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 192 A.8 The Group Oh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 48 A.9 The Group GS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 24 A.10The Group GP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 24 A.11The Group GK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 24 A.12The Group GS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 32 A.13The Group GK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 32 B Character tables of the considered discrete groups 121 B.1 Character Table of the Group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 1536 B.2 Character Table of the Group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 768 B.3 Character Table of the Group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 256 B.4 Character Table of the Group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 128 B.5 Character Table of the Group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 64 2 B.6 Character Table of the Group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 192 B.7 Character Table of the Group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 96 B.8 Character Table of the Group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 48 B.9 Character Table of the Group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 16 B.10 Character Table of the Group GS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 32 B.11 Character Table of the Group GP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 24 B.12 Character Table of the Group Oh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 48 B.13 Character Table of the Group GS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 24 C Classification of the momentum vectors and of the corresponding G irreps 140 1536 C.1 Classes of momentum vectors yielding orbits of length 6: {a,0,0} . . . . . . . . . . . . . . . . 140 C.2 Classes of momentum vectors yielding orbits of length 8: {a,a,a} . . . . . . . . . . . . . . . . 140 C.3 Classes of momentum vectors yielding orbits of length 12: {a,a,0} . . . . . . . . . . . . . . . . 141 C.4 Classes of momentum vectors yielding orbits of length 24: {a,a,b} . . . . . . . . . . . . . . . 141 C.5 Classes of momentum vectors yielding point orbits of length 24 and G representations of 1536 dimensions 48: {a,b,c} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 D Branching rules of G irreps 144 1536 D.1 Branching rules of the irreprs of dimension 1 and 2 . . . . . . . . . . . . . . . . . . . . . . . . 144 D.2 Branching rules of the irreps of dimensions 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 D.3 Branching rules of the irreps of dimensions 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 D.4 Branching rules of the irreps of dimension 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 D.5 Branching rules of the irreps of dimension 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 E Other relevant subgroups 157 (A,B,0) E.1 The Group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 128 (A,B,0) E.2 The Group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 64 (A,B,0) E.3 The Group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 32 (A,B,0) E.4 The Group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 16 (A,B,0) E.5 The Group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 8 (A,B,0) E.6 The Group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 4 (A,0,0) E.7 The Group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 256 (A,0,0) E.8 The Group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 128 (A,0,0) E.9 The Group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 64 (A,0,0) E.10 The Group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 32 (A,0,0) E.11 The Group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 16 (A,A,0) E.12 The Group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 32 F Some large formulae for the main text 183 III The Bibliography 185 3 Part I The Article 1 Introduction Classical hydrodynamics of ideal, incompressible, inviscid fluids subject to no external forces is described by Euler equation in three dimensional Euclidian space R3, namely by: ∂ u + u·∇u = −∇p ; ∇·u = 0 (1.1) ∂t where u = u(x, t) denotes the local velocity field and p(x) denotes the local pressure2. In vector notation eq. (1.1) reads as follows: ∂ ui + uj∂ ui = −∂ip ; ∂(cid:96)u = 0 (1.2) j (cid:96) ∂t and admits some straightforward rewriting that, notwithstanding the kinder-garden arithmetic involved in its derivation, is at the basis of several profound and momentous theoretical developments which have kept thecommunityofdynamicalsystemtheoristsbusyforalreadyfiftyyears[1,2,3,4,5,6,7,8,9,10,11,12,13]. With the present paper we aim at introducing into the classical field of mathematical fluid-mechanics a newgroup-theoreticalapproachthatallowsforamoresystematicclassificationandalgorithmicconstruction of the so named Beltrami flows, hopefully providing new insight into their properties. 1.1 Beltrami Flows and Arnold Theorem Let us then begin with the rewriting of eq.(1.2) which is the starting point of the entire adventure. The first step to be taken in our raising conceptual ladder is that of promoting the fluid trajectories, defined as the solutions of the following first order differential system3: d xi(t) = ui(x(t),t) (1.4) dt to smooth maps: S : R → M (1.5) t g from the time real line R to a smooth Riemannian manifold M endowed with a metric g. The classical t g case corresponds to M = R3, g (x) = δ , but any other Riemannian three-manifold might be used and ij ij there exist generalization also to higher dimensions. Adopting this point of view, the velocity field u(x, t) is turned into a time evolving vector field on M namely into a smooth family of sections of the tangent bundle TM: ∀t ∈ R : ui(x,t)∂ ≡ U(t) ∈ Γ(TM, M) (1.6) i 2Note that we have put the density ρ=1. 3Inmathematicalhydrodynamicspeopledistinguishtwonotions,thatoftrajectories,whicharethesolutionsofthedifferential equations (1.4) and that of streamlines. Streamlines are the instantaneous curves that at any time t = t admit the velocity 0 field ui(x,t ) as tangent vector. Introducing a new parameter τ, streamlines at time t , are the solutions of the differential 0 0 system: d xi(τ) = ui(x(τ),t ) (1.3) dτ 0 In the case of steady flows where the velocity field is independent from time, trajectories and streamlines coincide. Since we are mainly concerned with steady flows, in the present paper we often use the word streamlines and trajectories indifferently. 4 Next, using the Riemannian metric, which allows to raise and lower tensor indices, to any U(t) we can associate a family of sections of the cotangent bundle CTM defined by the following time evolving one- form: ∀t ∈ R : Ω[U](t) ≡ g ui(x,t)dxj ∈ Γ(CTM, M) (1.7) ij Utilizing the exterior differential and the contraction operator acting on differential forms, we can evaluate the Lie-derivative of the one-form Ω[U](t) along the vector field U. Applying definitions (see for instance [14], chapter five, page 120 of volume two) we obtain: (cid:16) (cid:17) L Ω[U](t) ≡ i ·dΩ[U] + d i ·Ω[U] U U U   = u(cid:96)∂ ui + gik∂ (cid:107) U (cid:107)2  g dxj (1.8)  (cid:96) k  ij (cid:124) (cid:123)(cid:122) (cid:125) gmnumun and Euler equation can be rewritten in either one of the following two equivalent index-free reformulations: − d(cid:0)p − 1 (cid:107) U (cid:107)2(cid:1) = ∂ Ω[U] + L Ω[U] (1.9) 2 t U or −d(cid:0)p + 1 (cid:107) U (cid:107)2(cid:1) = ∂ Ω[U] + i ·dΩ[U] (1.10) 2 t U Eq.(1.10) is one of the possible formulations of the classical Bernoulli theorem. Indeed from eq.(1.10) we immediately conclude that H = p + 1 (cid:107) U (cid:107)2 (1.11) 2 is constant along the trajectories defined by eq. (1.4). Turning matters around we can say that in steady flows, where ∂ U = 0, the fluid trajectories necessarily lay on the level surfaces H(x) = h ∈ R of the t function: H : M → R (1.12) defined by (1.11). Then if H(x) has a non trivial x-dependence it defines a natural foliation of the n- dimensional manifold M into a smooth family of (n−1)-manifolds (all diffeomorphic among themselves) corresponding to the level surfaces. Furthermore, as already advocated, the trajectories, i.e. the solutions of eq.(1.4), layonthesesurfaces. Inotherwordsthedynamicalsystemencodedineq.(1.4)iseffectively(n−1)- dimensional admitting H as an additional conserved hamiltonian. In the classical case n = 3 this means that the differential system (1.4) is actually two-dimensional, namely non chaotic and in some instances even integrable4. Consequently we reach the conclusion that no chaotic trajectories (or streamlines) can exist if H(x) has a non trivial x-dependence: the only window open for lagrangian chaos occurs when H is a constant function. Looking at eq.(1.10) we realize that in steady flows where ∂ Ω[U] = 0, the only open t window for chaotic trajectories is provided by velocity fields that satisfy the condition: i ·dΩ[U] = 0 (1.13) U This weak condition (1.13) is certainly satisfied if the velocity field U satisfies the strong condition: dΩ[U] = λ (cid:63) Ω[U] ⇔ (cid:63) dΩ[U] = λΩ[U] (1.14) g g 4Here we rely on a general result established by the theorem of Poincaré-Bendixson [15] on the limiting orbits of planar differential systems whose corolloray is generally accepted to establish that two-dimensional continuous systems cannot be chaotic. 5 where (cid:63) denotes the Hodge duality operator in the metric g: g (cid:63) Ω[U] = (cid:15) g(cid:96)kΩ[U]dxm ∧ dxn = u(cid:96)dxm ∧ dxn(cid:15) (1.15) g (cid:96)mn k (cid:96)mn (cid:63) dΩ[U] = (cid:15) gmpgnq∂ (g ur) dx(cid:96) (1.16) g (cid:96)mn p qr The heuristic argument which leads to consider velocity fields that satisfy Beltrami condition (1.14) as the unique steady candidates compatible with chaotic trajectories was transformed by Arnold into a rigorous theorem [1, 5] which, under the strong hypothesis that (M,g) is a closed, compact Riemannian three- manifold, states the following: Theorem 1.1 (Arnold) There are only two possibilities: a) Either the form Ω[U] is an eigenstate of the Beltrami operator (cid:63) d with a non vanishing eigenvalue λ (cid:54)= 0 g b) or the manifold M is subdivided into a finite collection of cells, each of which admits a foliation diffeomorphic to T2×R and every two-torus T2 is an invariant set with respect to the action of the velocity field U: in other words, all trajectories lay on some T2 immersed in the manifold M. Henceforth the desire to investigate the on-set of chaotic trajectories in steady flows of incompressible fluids motivatedtheinterestofthedynamicalsystemcommunityinBeltramivectorfieldsdefinedbythecondition (1.14). Furthermore, in view of the above powerful theorem proved by Arnold, the focus of attention concentrated on the rather unphysical, yet mathematically very interesting case of compact three-manifolds. Within this class, the most easily treatable case is that of flat compact manifolds without boundary, so that the most popular playground turned out to be the three torus T3. Reporting literally the words of Robert Ghrist in his very nice review [13]: on those occasions when compactness is desired and the complexities of boundary conditions are not, the fluid domain is usually taken to be a Euclidian T3 torus given by quotienting out Euclidian space R3 by the action of three mutually orthogonal translations. These slightly ironical words are meant to emphasize the main point which is outspokenly put forward by the same author few lines below: Since so little is known about the rigorous behavior of fluid flows, any methods which can be brought to bear to prove theorems about their behavior are of interest and potential use. Certainly most physical contexts for fluid dynamics do not correspond to the idealized situation of a motion in a compact manifold or, said differently, periodic boundary conditions are not the most appropriate to be applied either in a river, or in the atmosphere or in the charged plasmas environing a compact star, yet the message conveyed by Arnold theorem that Beltrami vector fields play a distinguished role in chaotic behavior is to be taken seriously into account and gives an important hint. Moreover, although boundary terms usually encode relevant physical phenomena, yet the history of periodic boundary conditions is a very rich and noble one in Quantum Mechanics, Classical Field Theory and also in Quantum Field Theory. It suffices to recall that periodic boundary conditions of quantized fields provide a formulation of finite temperature quantum field theory. In our opinion such arguments are a sufficient justification for the fifty year long efforts devoted by dozens of authors to the study of steady flows generated by Beltrami vector fields. On the other hand, what is somewhat surprising is that an overwhelming part of such efforts focused on a single example constructed on the T3 torus. The following vector field:   Ccos(2πy)+Asin(2πz) u(x,y,z) = V(ABC)(x,y,z) ≡  Acos(2πz)+Bsin(2πx)  (1.17)   Bcos(2πx)+Csin(2πy) which satisfies the Beltrami condition with eigenvalue λ = 1 and which contains three real parameters A,B,C defines what is known in the literature by the name of an ABC-flow (Arnold-Beltrami-Childress) [1, 2, 16] and during the last half century it was the target of fantastically numerous investigations. 6 Main aim of our work was to understand the principles underlying the construction of the ABC-flows, use systematically such principles to construct and classify all other Arnold-like Beltrami flows, deriving also, as a bonus, their hidden discrete symmetries. The issue of symmetries happens to be quite relevant at least in two respects. On one hand, in the case of the ABC model, it occurs that the choice of parameters (A : B : C = 1) which leads to the Beltrami vector field with the largest group of automorphisms leads also to the most extended distribution of chaotic trajectories. On the other hand symmetries of Beltrami flows have proved to be crucial in connection with their use in modeling magneto-hydrodynamic fast dynamos [17, 10, 11]. By this words it is understood the mechanism that in a steady flow of charged particles generates a large scale magnetic field whose magnitude might be exponentially increasing with time. No analytic results do exist on fast dynamos and all studies havebeensofarnumerical, yetwhiledealingwiththeselatter, crucialsimplificationsoccurandoptimization algorithms become available if the steady flow possesses a large enough group G of symmetries. In this case the magnetic field can be developed into irreducible representations of G and this facilitates the numerical determination of growing rates of different modes. It is important to stress that the linearized dynamo equations for the magnetic field B coincide with the linearized equations for perturbations around a steady flow. ThereforethesamedevelopmentofperturbationsintoirrepsofG isofgreatrelevancealsoforthestudy of fluid instabilities. In plasma physics Beltrami Flows are known under the name of Force–Free Magnetic Fields [9]. 1.2 The conception of contact structures Last but not least let us mention that Beltrami vector fields are intimately related with the mathematical conception of contact topology. This latter, vigorously developed in the last two decades starting from classi- calresultsofanalysisthatdatebacktoDarboux, GoursatandotherXIXcenturymaitres, isamathematical theory aiming at providing an intrinsic geometrical-topological characterization of non integrability, namely of the issues discussed above. As we have seen from our sketch of Arnold Theorem, the main obstacle to the onset of chaotic trajectories has a distinctive geometrical flavor: trajectories are necessarily ordered and non chaotic if the manifold where they take place has a foliated structure Σ ×R , the two dimensional h h level sets Σ being invariant under the action of the velocity vector field U. In this case each streamline h lays on some surface Σ . Equally adverse to chaotic trajectories is the case of gradient flows where there h is a foliation provided by the level sets of some function H(x) and the velocity field U = ∇H is just the gradient of H. In this case all trajectories are orthogonal to the leaves Σ of the foliation and their well h aligned tangent vectors are parallel to its normal vector. In conclusion in presence of a foliation we have the following decomposition of the tangent space to the manifold M at any point p ∈ M T M = T⊥Σ ⊕ T(cid:107)Σ (1.18) p p h p h and no chaotic trajectories are possible in a region S ⊂ M where U(p) ∈ T⊥Σ or U(p) ∈ T⊥Σ for p h p h ∀p ∈ S (see fig.1). This matter of fact motivates an attempt to capture the geometry of the bundle of subspaces orthogonal tothelinesofflowbyintroducinganintrinsictopologicalindicatorthatdistinguishesnecessarilynonchaotic flows from possibly chaotic ones. Let us first consider the extreme case of a gradient flow where Ω[U] = dH is an exact form. For such flows we have: Ω[U] ∧ dΩ[U] = Ω[U] ∧ ddH = 0 (1.19) (cid:124)(cid:123)(cid:122)(cid:125) =0 Secondly let us consider the opposite case where the velocity field U is orthogonal to a gradient vector field ∇H so that the integral curves of U lay on the level surfaces Σ . Furthermore let us assume that U is self h 7 Figure1: Schematic view of the foliation of a three dimensional manifold M. The family of two-dimensional surfaces Σ are typically the level sets H(x) = h of some function H : M → R. At each point of h (cid:107) p ∈ Σ ⊂ M the dashed vectors span the tangent space T Σ , while the solid vector span the normal space h p h to the surface T⊥Σ . Equally adverse to chaotic trajectories is the case where the velocity field U lies in p h T⊥Σ (gradient flow) or in T(cid:107)Σ p h p h similar on neighboring level surfaces. We can characterize this situation in a Riemannian manifold (M,g) by the following two conditions: i Ω[U] ⇔ g(U, ∇H) = 0 ; [U , ∇H] = 0 (1.20) ∇H The first of eq.s(1.20) is obvious. To grasp the second it is sufficient to introduce, in the neighborhood of any point p ∈ M a local coordinate system composed by (h,x(cid:107)) where h is the value of the function H and x(cid:107) denotes some local coordinate system on the level set Σ . The situation we have described corresponds h to assuming that: U (cid:39) U(cid:107)(x(cid:107))∂ ; ∂ U(cid:107)(x(cid:107)) = 0 (1.21) (cid:107) h Under the conditions spelled out in eq.(1.20) we can easily prove that: i dΩ[U] = 0 (1.22) ∇H Indeed from the definition of the Lie derivative we obtain:   i∇HdΩ[U] = L∇HΩ[U] −di∇HΩ[U] (1.23) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) =Ω[[U,∇H]]=0 =0 Since we have both i Ω[U] = 0 and [U , ∇H] = 0 it follows that also in this case: ∇H Ω[U] ∧ dΩ[U] = 0 (1.24) 8 Figure 2: A schematic picture of a contact structure. Given the Reeb field U, associated with a contact one-form α, we can consider the family of hyperplanes orthogonal, at each point of the manifold, to the vector field U. These hyperplanes are defined as the kernel of the contact form α and constitute the contact structure. Indeed the three-form Ω[U] ∧ dΩ[U] has no projection in the direction ∇H and therefore it lives on the two dimensional surfaces Σ : but in two dimensions any three-form necessarily vanishes. This heuristic h arguments motivate the notion of contact form and contact structure that capture the non integrability of a vector field in a topological, metric independent way. Definition 1.1 Let M be smooth three-manifold. A contact form α ∈ Γ(CTM,M) is a one-form such that α ∧ dα (cid:54)= 0 (1.25) Definition 1.2 Let M be smooth three-manifold and α a contact form on it. The rank two vector-bundle of all vector fields X which satisfy the condition: i α = 0 (1.26) X is named the contact structure CS defined by α. α In view of what we discussed above it is clear that the definition of contact structures captures the notion of maximal non-integrability. At each point p ∈ M the contact structure is a two-dimensional plane singled out in the tangent space T M by the condition (1.26). This smooth family of planes, however, cannot be p considered as the tangent plane of the level surfaces of any foliation. This is ruled out by the condition (1.25). Let us next introduce the notion of Reeb-like field Definition 1.3 (Reeb-like field). Let M be a smooth three-manifold and α a contact one-form defining a contact-structure. A Reeb–like field for α is a vector field U satisfying the following two conditions: i α > 0 ; i dα = 0 (1.27) U U 9