A Lie Algebra version of the Classical or Quantum Hamiltonian Perturbation Theory and Hamiltonian Control with Examples in Plasma Physics

Transcription

1 A Lie Algebra version of the Classical or Quantum Hamiltonian Perturbation Theory and Hamiltonian Control with Examples in Plasma Physics Michel VITTOT Center for Theoretical Physics (CPT), CNRS - Luminy Marseille, France vittot@cpt.univ-mrs.fr

2 Warm Thanks to the Organizers! Parts of this work done in collaboration with: Ricardo LIMA, Cristel CHANDRE, Xavier LEONCINI, Natalia TRONKO, Pierre DUCLOS (CPT, Marseille) And for the applications to plasma physics: Guido CIRAOLO (MSNM, Marseille) Fabrice DOVEIL, Alessandro MACOR & Yves ELSKENS (PIIM, Marseille) Philippe GHENDRIH, Xavier GARBET, Yanick SARAZIN (CEA-DRFC, Cadarache) Marco PETTINI (Osservatorio Astrofisico di Arcetri, Firenze, Italy, and now: CPT) Alain BRIZARD (St-Michael College, Colchester, Vermont, USA) 2/73

3

4 Classical/Quantum Hamiltonian Mechanics & Perturbation Theory Topics: - The algebraic ( Heisenberg ) point of view simplifies & unifies the computations - Which property to preserve? -Amethod for an Hamiltonian Control - Applications in Plasma Physics - KAM Theory without integrability & compact form of Hamilton-Jacobi equation - Ex: Auto-Consistent Interaction between Particles & Electromagnetic Fields - Ex: Propagator & Floquet Decomposition - Great number of degrees of freedom - Applications in Quantum Mechanics - Extension of the Lagrange Inversion Formula - Projects 4/73

5 Motivations Problem: perturbation theory of a non integrable H? Which property to preserve after perturbation? Ex: H integrable. We want to restore this integrability after perturbation. Or at least, an invariant torus. But if H is non integrable, which structure to preserve? KAM without actions-angles? Ex: non-canonical Poisson bracket: in field theory, hydrodynamics... Another motivation: to get a compact version of the Hamilton-Jacobi equation in order to use some inversion theorems. An answer: a sub-lie algebra U of the Lie-algebra V of Observables. In general dim(v) =. But the flow on V is linearized. This is the Heisenberg point of view. 5/73

6 Phase Space 6/73

7 Algebra 7/73

8 Hamiltonian Structure : V is a Lie Algebra = vector space with a linear application {..} : V L := L(V, V) from V into its linear endomorphisms L, s.t: V,W V {V } W = {W } V and the Jacobi identity: V,W V {V }{W } = {{V } W } + {W }{V } Ex: Classical Mechanics: V = C (Φ, R) with Φ = Poisson (or Symplectic) Manifold, the Phases Space. The Lie bracket = Poisson bracket:... : T Φ T Φ {V } W := dv(dw) Ex: Quantum Mechanics: V = B(Ψ, Ψ) where Ψ = Hilbert Space. The Lie bracket = commutator (divided by i ). 8/73

9 We set H := ad H = the operator bracket with H. It is an endomorphism of V,a matrix. Its flow is e t H i.e the evolution at time t of any observable V is: V t := e t H V i.e. Vt = H V t i.e the Hamilton s equation. Ex: Classical Mechanics: Φ = set of p, q. Then V t (p, q) =V (p t,q t ) where p t,q t = usual motion in Phases Space Φ. Hence e t H V = V o e t H with H = dh (the Hamiltonian vector field of H ). Linearization of the flow! Cf. von Neuman, Koopman. Ex: Quantum Mechanics: e t H V = e i th/ V e i th/ 9/73

10 So let H be in a sub-lie Algebra U (denoted admissible ) of the Lie Algebra V of observables. Ex: U may be the set of Hamiltonians which preserve a given torus. Or those which admit some given constants of motion. Ex: the observables which commute with the angular momentum. Or with a given spectral projector. For aperturbation V quite general, but small of H, wewant an additive control term F (for instance quadratic in V ) for H := H + V s.t exists a sub-lie algebra Ũ isomorphic to U which contains the perturbed controlled system H := H +F.Wewant the Lie-isomorphism ( change of variables ) which conjugate U and Ũ. The isomorphism is crucial: so we will have the same properties as for H. A more difficult problem is to calculate this isomorphism without modification of H : KAM theory. So: H = H.Wewill need an even smaller V. 10/73

11 Sub-Algebra=U 11/73

12 Control 12/73

13 KAM 13/73

14 Ex 1: U = the observables which preserve Ker P with P := {P } and P Ker H, i.e is a Constant of Motion of H. Indeed P t = e t H P = P Ex: P = H. The associated perturbation theory is named global. Ex 2: U = the observables which preserve Ker M with M = M 2 a projector. Ex: when V = C (Φ, R). Choice: M = composition with a projection M :Φ Φ, i.e for any observable V : M V = V o M with M = M o M Ex: M (p, q) =(p 0,q) for some p 0. Hence U = the observables which preserve the initial condition p 0, i.e which evolve any V vanishing on (p 0,q),into a V t vanishing on (p 0,q), for any t. Saying H U means that we know many trajectories of H. The equivalent of an invariant torus. Idem in quantum mechanics: M = some projector. The associated perturbation theory is named local. 14/73

15 N.B: H D the Lie derivations of V, i.e.: H ad V = ad H V +ad V H. The flow is a Lie morphism, like any exponential of derivation, i.e: e H {V } = {e H V } e H Canonical Change of Variables, or Unitary Transform. Let V be aperturbation of H.Ingeneral H := H + V / U. We want to isomorphically deform U into Ũ (and to compute some small F ) s.t. H := H + F Ũ,inorder to restore the property of H. 15/73

16 A version of Hamiltonian Control Let H + B = the perturbed, controlled Hamiltonian, and transformed by the automorphism: If S verifies: then : B := V H S U and B = O(V ) e S ( H + F ) = H + B U i.e. e t ( H + F) = e S.e t (H + B).e S with F := ( e S 1 e S S ) B + 1 S e S S V = S B + V = O(V ) 2 The fraction is given by a series or integral: n=0 ( S) n = 1 (n+1)! 0 The notation {V } =ad V really simplifies the writings! Ex: F := 1 { e {S} 1 e {H} S + e {S} S B V = {S} S dt e t S H S + e S B V } 16/73

17 Crucial Consequences Ex 1: When U is the set of observables which preserve Ker P with P Ker H, then this theorem insures that H also has a Constant of Motion P := e S P. Ex 2: When V = C (Φ, R),and when U is the set of observables which preserve Ker M where M is the composition with a projector M of phases space, this theorem insures that H also has a whole set of trajectories, associated to Ker M, with M := e S M e S is isomorphic to M and is a projector defined by the composition with M isomorphic to M. Ex 3: When V = B(Ψ, Ψ), with Ψ = Hilbert Space, and when U = Ker P, for some projector P, this theorem insures that H also has an eigen-projector P := e S P, isomorphic to P. 17/73

18 Control of the forced pendulum H = A2 2 V = ε (cos θ + cos(θ t)) Choose U = the observables which preserve Ker M, with M the projector on an initial condition A 0 =Ω/ {0, 1}. Then the control term: F = ε2 2 ( cos θ Ω ) 2 cos(θ t) + Ω 1 restores the invariant torus passing through A 0 =Ω (for any θ 0 ) which existed when ε =0. This control is useful only if ε is above the breaking value of the torus. Otherwise the KAM theorem insures the existence of the perturbed torus. This control will be small only if ε < 1 (approximately) otherwise ε 2 > ε. Poincaré Sections for Ω = 0.38 : 18/73

19 Forced Pendulum - epsilon = /73

20 Forced Pendulum + Control: epsilon = /73

21 Forced Pendulum + Control: epsilon = With initial conditions above 21/73

22 Applications in Plasma Physics Published in: Phys. Rev. Letters, Nonlinearity, Physica D,... 22/73

23 Travelling Wave Tube Electrons beam & gun, velocity analyzer, antenna, confining magnetic coil 23/73

24 Velocity Distribution of entering electrons (10^6 m/s) 24/73

25 Velocity Distribution of exiting electrons (10^6 m/s) 25/73

26 Velocity Distribution of exiting electrons With a control, of energy = 1% of total 26/73

27 27/73

28 ITER 28/73

29 29/73

30 30/73

31 31/73

32 32/73

33 33/73

34 34/73

35 35/73

36 Control of a simple model: ExB A simple model of a Tokamak: no retro-action of the plasma on the fields. The phase space is a toroidal section (orthogonal to B), with coordinates x, y treated as canonically conjugated. x = radius, y = poloidal angle. B is taken constant, uniform and strong. The hamiltonian is the electric potential V (x, y, t). The coupling constant ε is the inverse of the B. We treat the toroidal angle as a time. for some random phases φ n,m V (x, y, t) =ε N n,m=1 cos (nx + my + φ n,m t) (n 2 + m 2 ) 3/2 We plot the diffusion coefficient which shows a decrease by 1 or 2 order of magnitude after our global control method is (numerically) implemented by a small change of the external potential V,oforder ε 2. Cf: Phys Rev E 69 (2004) /73

37 ExB model: Diffusion Coefficient, without & with control 37/73

38 ExB model: Diffusion Coefficient, with delta*control 38/73

39 Local control: choose a radius x o where to build a transport barrier Apossible control adapted to x 0 is: Ṽ V + F V (x + y f(y, τ),y,t) where f(y, t) t 0 V (x 0,y,τ)dτ 39/73

40 Figure 1: Phase portraits without control and with the exact control term, for ε = 0.9, x 0 =2,N traj = /73

41 Figure 2: Phase portraits without control and with the truncated control term, ε = 0.3, T = 2000, N traj =50 41/73

42 Control of the magnetic lines We treat 2 examples, using the Clebsh representation of the magnetic field, with some hamiltonian H(ψ, θ, ϕ) representing the poloidal flux, with ψ the normalised toroidal flux (proportional to the square of the radial coordinates: ψ = 1 being the boundary of the plasma), and θ, ϕ the poloidal and toroidal angles. The first example describes some coupled tearing modes, with H quartic in ψ and perturbed by 2 harmonics: cos(2θ ϕ)+ cos(3θ 2ϕ). The second example describes some ergodic divertor, with the same quartic term in ψ but with a more complex perturbation. Cf: Nuclear Fusion, 46 (2006) p 33 This method is also in Ali & Punjabi: Plasma Phys. & Controlled Fusion 49 (2007) p /73

43 Tokamak: B Lines, without Control 43/73

44 Tokamak: B Lines, with Control. Initial conditions below 44/73

45 Tokamak: B Lines, with Control. Initial conditions above 45/73

46 Tokamak Divertor: B Lines, without Control 46/73

47 Tokamak Divertor: B Lines, with Control. Initial conditions below 47/73

48 Tokamak Divertor: B Lines, with Simplified Control. Initial conditions below 48/73

49 Perturbation Theory What to do when we can t (or don t want to) modify H? Usual Hypothesis on H : integrability. Actions-Angles coordinates. What to do in other cases? Ex: if the bracket is non-canonical? Ex: if the phases space is not of even dimension? We assume that H has a left inverse G, another endomorphism, (matrix) of V into itself, restricted to U : G H G = G Ker G U This hypothesis is sufficient to write a perturbation theory for H without Actions-Angles. 49/73

50 When H is integrable (in classical mechanics), G is easy. There exists coordinates (A, θ) s.t. H = H(A, θ) is independent of θ. Local Case: Taylor in A around A 0 U as in Ex 2 with M when the denominator 0. Else G gives 0. (choice A 0 =0). Then H(A) =ΩA + O(A) 2. And associated to A 0 =0. And G is defined, via Fourier (θ ν) by: (G V )(A, ν) := V (0,ν) i Ω ν When H is diagonalizable (in quantum mechanics), G is easy. H = n h n P n for some eigen-values h n and eigen-projectors P n. Let us define V n,m := P n VP m. And G is the division by the spectral gaps of H. when h n h m. Else G gives 0. (G V ) n,m := i h n V n,m h m 50/73

51 When G exists, we set N := H G, R := 1 N (= projectors) and Γ := {G...} sends V into the Lie derivations. Theorem (KAM type): If we have a solution W Rg N of the Hamilton-Jacobi equation: N eγw 1 ΓW W = N eγw V Then H Ũ where: Ũ := e ΓW U is a sub-lie algebra of V, isomorphic to U. e ΓW = Lie-isomorphism, Canonical Change of Variables, or Unitary Transform. NB: Γ is independent of or of any scaling of the Lie bracket, since Γ = {G...} and G = (left) inverse of {H} There also exists a version with t Z ( Mappings ). 51/73

52 We have rigourous estimates on the size and the regularity of V and on the Diophantine condition on G in order to have a solution to the Hamilton-Jacobi equation. For ex: when we are given a sequence of norms on V (Frechet structure). We find : W = N=1 W 3 = N(ΓV ) 3R 1 6 W N with W 1 = N V, W 2 = N ΓV R +1 2 (ΓV ) R +1 2 V + N [ ( Γ (ΓV ) R +1 2 )] R +1 V 2 V,... V Usual Tools: Nash-Moser regularization, Newton-Raphson method... This theorem may be rewritten as: e ΓW H = H + B with B := e ΓW V eγw 1 ΓW W Ker N U This is the Normal Form of H relatively to U. Higher Order Control: if W is truncated, it yields a control F of order O(V n ). 52/73

53 Non-Resonant Integrability An important Class of integrable H : the non-resonant ones. A possible definition: Iff its constants of motion Ker H form an abelian and maximal sub-lie algebra of V. The abelianity (the bracket vanishes) and the maximality (any algebra which strictly contains Ker H is non abelian) come from the involution of the constants of motion, and that there must exists a lot of them. Ex: with (A, θ) R L T L, let H = Ω.A i.e H = Ω. θ Ker H will be maximal abelian iff: Ω Z L = {0} Then Ker H = the observables independent of θ, which are indeed in involution. On contrary, ex: Ω 1 = 0 : then Ker H contains the observables which depend on A and θ 1 only, which don t commute with the other functions of A 1. 53/73

54 Auto-Consistent Interaction between Particles & Electromagnetic Fields An Hamiltonian approach for the Maxwell-Vlasov equations V = the real functions of (f,e,b) with f = f α (p, q) = Vlasov density of particles of type α at the position q R 3 and with the momentum p R 3 Ex: α {+, } :positive ions & electrons. Charge = e α. Mass = m α > 0 E = E(q),B= B(q) = the fields R 3 V is a Lie algebra with the Maxwell-Vlasov bracket (!), for any observables V = V (f,e,b) and W = W (f,e,b) (with D = p := q ): {V }W = [ ] d 3 q ( B V ) E W E V ( B W ) + α d 3 qd 3 pf α [(D fα V ) fα W ( fα V )D fα W +e α ( E V )D fα W e α (D fα V ) E W e α B (D fα V ) D fα W ( ) ] 54/73

55 This Bracket is non canonical, i.e. not of the form: {V } W := ( A V ) θ W ( θ V ) A W The Hamiltonian: H = 1 2 d 3 qb 2 + α d 3 qd 3 pf α γ α with γ α := (m 2 α + p 2 ) 1 2 V = 1 2 d 3 qe 2 Hence H := H + V is the sum of the kinetic energy of the particles and of the Poynting energy. No coupling constant! Choice: U = the observables which preserve Ker H Cf: Pauli - Bialynicki-Birula - Morisson - Marsden - Weinstein 55/73

56 The Hamilton equations give the 2 Maxwell and the Vlasov equations: Ė := HE = B α e α d 3 pf α v α Ḃ := HB = E f := Hf = v α f α e α (E + v α B) Df α where v α := p/γ α is the velocity, of modulus <c (= 1). The 2 other Maxwell equations are constraints on initial conditions. evolution of H or H :.B =0.E α e α d 3 pf α =0 Preserved by the NB: this model is gauge invariant. The Poincaré group is included in the Lie-isomorphisms: relativistic covariance. 56/73

57 The simple flow of H The flow of H leaves B invariant and moves the matter in a simple cyclotronic way: B t = B 0, f α,t (q, p) =f α,0 (q α,t,p α,t ) where q α,t,p α,t is the usual cyclotronic motion in the field B, without any E, and the motion of E is: We have explicitly: q α,t = q α,0 + t E t (q) =E 0 (q) + t ( B)(q) e α dτ t 0 dτ p α,t /γ α, p α,t = T e ( 0 ) t e α dτ B τ /γ α 0 dp f α,τ (q, p) p/γ α p α,0, γ α = γ α,t = γ α,0 with an ordered exponential, and with B τ := B(q τ )....Aproduct of rotations. So our goal is to deform this trajectory when we add the perturbation V = E 2 /2. And to determine some initial conditions (f 0,E 0,B 0 ) which still insure some confinement... 57/73

58 Propagator and Floquet Decomposition Third ex: a morphism-projector. Let M be a projector of V into V, which is also a Lie-morphism. Let Y := Rg M = Ker (1 M) V. And U := {B V s.t. B Y Y} NB: not necessary that H be an interior derivation, i.e that H = {H} for some H. Hypothesis: H D and H M = 0 i.e H Y Y And H has a left inverse G relative to M, i.e. s.t: G H G = G Ker G = Y We recall the notations: Γ = {G...}, and N := H G, R := 1 N. We have: Rg R = Ker N = Y Let M t := M e t H = morphism-projector Forany perturbation V V,westill have the KAM-type theorem, for H := H + V 58/73

59 If we have a solution W Rg N = Ker R of: N B = 0 i.e. B Y with B := e ΓW V eγw 1 ΓW then we still get the KAM result and: W e {ΓW } H := e ΓW H e ΓW = H + B with B = {B} B Furthermore t, s R : M t e (t s) H = e W t,s M s e W t,s := e { M t G W } e (t s) B e {M s G W } If we set t, s R : Y Y Y s (t) := e W t,s Y then Y s (t) verifies the evolution equation: Ẏ s (t) = V t Y s (t) and Y s (s) = Y with V t := {M t V } 59/73

60 Hence e W t,s is the propagator associated to V t = M t V,between the dates s and t. The Floquet decomposition of the propagator: conjugates the flow of H in the large algebra V with the flow of W t,s in the small algebra Y. Ex: Time Dependent Hamiltonians Let Y = an arbitrary Lie-algebra, which we consider as time independent observables. Let us introduce a time dependence τ R (or sometime τ Z ) byextending Y into V := C (R, Y) For quasi-periodic dependence in τ,wemay also define V via Fourier, by V := l 1 (K, Y) with K an additive sub-group of R, countable. Ex: if the dependence in t is periodic, then K = Z,orifitis quasi-periodic with 2 frequencies ω 1,ω 2 (of irrational ratio) then K = ω 1 Z + ω 2 Z R. The number of frequencies = its dim as a Z-module, i.e vector space on the ring Z. We still have a Lie-algebra with the induced bracket. 60/73

61 A time-dependent observable V V generates an autonomized flow in V when we add to it H := τ or in Fourier H := i k. These are not interior derivations. Hence an autonomized flow in V is of the type H = H + V An example of morphism-projector M is the composition with the projector R R which sends an arbitrary date t onto 0: V V τ R (M V )(τ) := V (0) or V V k K (M V )(k) := then HM = 0 with H = τ,or H := i k l K V (l) δ 0 (k) We may define the left inverse G in Fourier by: V V k K \{0} (G V )(k) := V (k) i k 61/73

62 Great Numbers of Degrees of Freedom Let L be the Number of Degrees of Freedom. And ε L be the maximal size permitted for the perturbation V,below which the KAM theorem applies: ε L 0,ifL. The original KAM theorems gave, with analytic perturbations: L log L ε L e We have improved (in 1985) this behavior ε L into: (log L)2 ε L e And we have constructed 2 classes of perturbations for which ε L is independent of L. This was afirst extension of the KAM theorem to the case of an infinite number of degrees of freedom. 62/73

63 Quantum KAM theory Let H be aquantum Hamiltonian (an infinite matrix) diagonal with dense spectrum, perturbed by some V slowly decreasing with the distance to the diagonal. We want the eigen-projectors (and eigen-values) associated to H := H + V. Time-Dependent case: H := i ω τ + K where K = n h n P n with h n R and P n are orthogonal projectors: If σ>0 s.t. n N (n) σ < with (n) := inf m N\{n} h m h n and if V (t) isofclass C σ+1/2 and small, then there exists a Cantor set of values of ω, of non-zero measure, s.t. the perturbed projectors Pn are isomorphic to P n. Ex: h n = n α with α>1 and σ = 1/(α 1) 63/73

64 Extension of the Lagrange Inversion Formula We want a fixed-point W = G(W ) with G : V V (topological vector space) and G(W )= n 0 Ĝ(n)W n converges around 0. Ĝ(n) = completely symmetric n-linear application, of V n into V. Then: W = Ĝ(ν N 1 ) Ĝ(ν N 2)... Ĝ(ν 1 ) Ĝ(ν 0) N 1 ν T (N) { with : T (N) := ν =(ν 0,ν 1,...,ν N 1 ) {0, 1,...,N 1} N s.t. : k {0, 1,...,N 1} : ν k := ν 0 + ν ν k k and ν N 1 = N 1 Hence : W = Ĝ(0) + Ĝ(1) Ĝ(0) + (Ĝ(1) Ĝ(1) Ĝ(0) + Ĝ(2) Ĝ(0) Ĝ(0) ) Ex: Baker-Campbell-Hausdorf, Lindstedt series,... } +... Ex: W = G(W ):=Z.eȲ W for some Z V and Ȳ V := L(V, R). A solution is: W = Z.g(Ȳ Z) with g(x) (n +1) n 1 := n 1 n! = g(x) converges if x e 1 (but may be extended if x< e 1 ). x n 64/73

65 Rooted, Acyclic, Planar Graphs 65/73

66 Projects Gyrokinetic description of fusion plasmas: ITER Control of the confinement of these plasmas Control of Free Electrons Lasers: generation of harmonics Continue the Quantum Formulation in terms of Lie-algebras, of the Heisenberg point of view: Projectors, Resolvents, Quantum States (pure, entangled...) Quantum Information: control of the decoherence. Rydberg Atoms: change of behavior: classical - quantum Other applications of the Lagrange Inversion Formula 66/73

67 Averaging: large frequency limit Adiabatic Regularization Hydrodynamics (formulation similar to Maxwell-Vlasov) New proof of the KAM theorem, via Frechet structure or via the method of Giorgilli-Locatelli New proof of the quantum KAM theorem: decreasing gaps 67/73

68 And perhaps also: Lie-algebra formulation of the Hamiltonian General Relativity (Ashtekar, Rovelli) Lie-algebra formulation of the complete integrability: bi-hamiltonian structure, Yang- Baxter Equation, Lax Pair. PDE: KdV,... Mappings and Symplectic Integrators To increase the list of basic canonical transforms Control of Ions Traps Bose-Einstein Condensation Extension to open systems: Lindblad, Nose-Hoover 68/73

69 Some of our papers... 69/73

70 [1] J. BELLISSARD, M. VITTOT: Heisenberg s picture and non commutative geometry of the semi classical limit in quantum mechanics, Annales de l Institut Henri Poincare, 52, 3,(1990) [2] P. DUCLOS, P. STOVICEK, M. VITTOT: Perturbation of an eigen-value from a dense point spectrum: an example. J. Phys. A: Math. Gen. 30, (1997) [3] P. DUCLOS, P. STOVICEK, M. VITTOT: Perturbation of an eigen-value from a dense point spectrum: a general Floquet Hamiltonian. Annales de l Institut Henri Poincare, 71, 3,(1999) p 241. [4] P. DUCLOS, O. LEV, P. STOVICEK, M. VITTOT: Weakly regular Floquet Hamiltonians with pure point spectrum, Reviews in Mathematical Physics, 146 (2002), p /73

71 [5] M. VITTOT, C. CHANDRE, G. CIRAOLO, R. LIMA: Localised control for nonresonant Hamiltonian systems, Nonlinearity 18, (2005) [6] T. BENZEKRI, C. CHANDRE, X. LEONCINI, R. LIMA, M. VITTOT: Chaotic advection and targeted mixing, Phys. Rev. Letters 96, (2006). [7] M. VITTOT: Perturbation Theory and Control in Classical or Quantum Mechanics by an Inversion Formula, J. Phys. A: Math. Gen. 37, (2004) p [8] C. CHANDRE, G. CIRAOLO, F. DOVEIL, R. LIMA, A. MACOR, M. VITTOT: Channelling chaos by building barriers, Phys. Rev. Letters 94, (2005). [9] C. CHANDRE, M. VITTOT, G. CIRAOLO, Ph. GHENDRIH, R. LIMA: Control of stochasticity in magnetic field lines, Nuclear Fusion 46 (2005) p [10] C. CHANDRE, M. VITTOT, Y. ELSKENS, G. CIRAOLO, M. PETTINI: Controlling chaos in area-preserving maps, Physica D 208, (2005) p /73

72 [11] M. VITTOT: Perturbation or Control of a Non-Integrable Classical or Quantum Hamiltonian with Examples in Plasma Physics. Preprint CPT (2007). [12] M. VITTOT, N. TRONKO: Perturbation Theory for the Self-Consistent Interaction of a Plasma and Electromagnetic Fields. Preprint CPT (2007). 72/73

73 Thank you! 73/73