Hamiltonian formulation of reduced Vlasov-Maxell equations Cristel CHANDRE Centre de Physique Théorique CNRS, Marseille, France Contact: chandre@cpt.univ-mrs.fr
importance of stability vs instability in devices involving a large number of charged particles interacting ith fields: plasma physics (tokamaks), free electron lasers Here: reduced models of such systems (easier simulation, better understanding of the dynamics) Outline - Hamiltonian description of charges particles and electromagnetic fields - Reduction of Vlasov-Maxell equations using Lie transforms Alain J. BRIZARD (Saint Michael s College, Vermont, USA) - Reduced Hamiltonian model for the Free Electron Laser Romain BACHELARD (Synchrotron Soleil, Paris) Michel VITTOT (CPT, Marseille)
Motion of a charged particle in electromagnetic fields > in canonical form e p A( q, t) c f g f g H( pq,, t) = + ev( q, t) ith { f, g} = m q p p q H p = { p, H} = equations of motion : q H q = { q, H} = p > in non-canonical form: physical variables 1 e v = (, t) m p A q c x = q 1 1 (,, ) (, ) {, } f g f g eb f g H vxt = mv + ev xt ith f g = + m x v v x mc v v x = { x, H} = v gyroscopic bracket equations of motion : e {, } = v B v = v H + E m c
Definition: Hamiltonian system - a scalar function H, the Hamiltonian { FG} { FG} = { GF} { F GK} = { F G} K + G{ F K} { } - a Poisson bracket, ith the properties antisymmetric,, Leibnitz la,,, { FG K} {{ KF} G} {{ GK} F} Jacobi identity,, +,, +,, = 0 - equations of motion df = dt { FH, } - a conserved quantity { FH}, = 0
Eulerian version: case of a density of charged particles ( xvt) - density of particles in phase space f,, 1 example: f ( xv,, t) = δ( x x() t ) δ( v v() t ) Klimontovitch distribution i i N i - evolution given by the Vlasov equation f e f f v = v + t m E B c v - Eulerian, not Lagrangian: for any observable d f F F F, e have = =, dt t F H - still a Hamiltonian system Hamiltonian : H f = m dxdvf v 3 3 + ev = δf 3 3 ith F, G dxdvf, δg
Eulerian version: case of a density of charged particles ( ) 3 - an example: ρ f x = e d v f( x, v) 0 0 ρ 3 3, δρ δ dxdvf H = ρ =, t H - functional derivatives f = f δf F +φ F + d xd v φ+ O φ ( ) 3 3 δρ δh 1 - here: = δ( x x ) and = v + e m ev 0 - therefore: ρ = t = x 3 J ith J f e d v f v
Vlasov-Maxell equations: self-consistent dynamics > description of the dynamics of a collisionless plasma (lo density) Variables: particle density f(x,v,t), electric field E(x,t), magnetic field B(x,t) f e f f v = v t m E+ B c v B = c E t E = c B 4 π J t here E = 4πρ and B = 0
Vlasov-Maxell equations... still a Hamiltonian system 3 3 m 3 Hamiltonian : H,, f E B = dxdvf v + dx 3 3 δ δ, dxdvf F G ith, F G = 4πe 3 3 f dxdv δf δg δf δg + m v δ δ E E 3 δ δ δ δ + 4πc d x F G F G δ δ δ δ E B B E E + 8π B d,, f F F Equation of motion for F = =, E B : dt t F H 3 Remark: div B et div E dpf are conserved quantities antisymmetry, Leibnitz, Jacobi Morrison, PLA (1980) Marsden, Weinstein, Physica D (198)
From microscopic to macroscopic Vlasov-Maxell equations - Elimination (or decoupling) of fast time and small spatial scales for a better understanding of complex plasma phenomena - reduced Maxell equations in terms of D and D = 4πρR D = c H 4 π JR t D = E + 4 π P, H = B 4 π M here P ρ = ρ+ P, J = J c M R R t reduced polarization density / magnetization current / polarization current density H - Can e represent the reduced Vlasov-Maxell equations as a Hamiltonian system? Hint: use of Lie transforms - Deliverables: Expressions of the polarization P and magnetization M vectors
Reduced fields as Lie transforms of f, E and B ( t) ( t) f ( t) Given a functional S,,,,,, Ex Bx xv, e define some ne fields as 1, +,, + E S E S S E D E L 1 = e =, +,, H S B B S B + S S B F f f 1 S, f,, f + + S S Remark: If the variable χ is only a function of x L S then e χ is only a function of x The functionals transforms into F e F, resulting in a ne Hamiltonian and a ne Poisson bracket... L = S
Polarization, magnetization, reduced density, etc 1 ( ) e L δs e 1 3 c d vf δs S P = E = + 4π δ m f B v δ 1 ( L 1 e ) δs S M = B = c + 4π δe so that D= E+ 4πP H = B 4πM F Reduced evolution operator L L S S e e t t F L = e L L S S S e,e =, F H F H
Reduced Vlasov-Maxell equations D = c H 4 π J t H = c D t D D = +, c 4 R t t D H H = H π J H H = +, M = c 4π + 4πc t t H H H D P t F e F Reduced Vlasov equation F v = v t m D+ H c v δ 4πe f δ F = f S f, S + m v δe guiding center theory / gyrokinetics
What S? D E L e = H S B F f - Elimination of small spatial and fast time (averaging) scales of f(x,v,t) : guiding-center collisionless plasmas (lo frequency phenomena) gyrokinetics reduced models for free electron lasers -Use of KAM algorithms (at least one step process) For F = f +, homological equation P( δ f +, f ) S = 0 - Advantages: preserve the structure of the equations, invertible, symbolic calculus Brizard, Hahm, Rev. Mod. Phys. (007)
Strategies to reduce Vlasov-Maxell equations > rigorous: H e L = S H > non-rigorous: truncate the Hamiltonian system - the equations of motion - the Hamiltonian and the Poisson bracket > the canonical version provides a ay out...
Reduced model for the Free Electron Laser From: Vlasov-Maxell Hamiltonian 3 3 ( xp, ) 1+ p H, E B = + 3, f dxdpf dx E + B To: Bonifacio s reduced FEL Hamiltonian model p HfI,, ddpf(, p ) Isin( ) ϕ= θ θ + θ ϕ ith intensity and phase of the electromagnetic ave ( I, ϕ) in a Hamiltonianay
A Free Electron hat? λ λ (1 ) rms γ u LEL = + K
Vlasov-Maxell: canonical version Change of va riables: ( f EB) ( xp, ) = f xp, + A( x) f m E = Y B = A Bracket: canonical F G = Hamiltonian: ( ) 3 3, dxdpfm,, ( f, YA, ) δf δg δf δg 3 δ δ δ δ + d x F G F G δ δ δ δ p p A Y Y A m m m m H = 1 + ( p A) 3 3 dxdpfm m + 3 dx Y + A Translation of A by a constant function (external field-undulator): A x A (, t) ( x) + A( x, t) Bracket: canonical (canonical transformation) Hamiltonia n: H ( p A A) 3 3 = dxdpf 1 + + m 3 dx Y + A A+ A a A helicoidal undulator: A = e+ e ik z ik z * ( e ˆ e ˆ )
One mode for the radiated field ŷ ˆx z Paraxial approximation and circularly polarized radiated field i ikz * ikz * xˆ+ iyˆ A = ( ae ˆe a e ˆe ) here ˆe = k ikz * ikz * Y = ( ae ˆe+ a e ˆe ) Remark: a does not depend on x and y but depends on time (dynamical variable) Bracket:, FG = Hamiltonian: H + L: interaction length V: interaction volume 3 3 dxdpfm ik δf δg δf δg F G F G + * * V a a a a p p m m m m ikz ikz ( ˆ a ˆ ) 3 3 * * * = dxdpf + + aa i a m iks * ikz * kvaa dz a a 1 p e e e e A + A ikz ( e ˆe e ˆ e * ) A
Dimensional reduction The fields do not depend on x and y no transverse velocity dispersion f ( xp, ) = f ( x, p ) δ ( p ) if p ( t = 0) = 0 If p then no modification of the distribution Bracket: () t = 0 ( x, y) ( xp) ( ) ( p ) ( ) ( ) ( ) ( ) f, = f z, p δ δ x δ y if x t = 0 = y t = 0 = 0 (injection at the center) Hamiltonian: H = ik, δ δ δ δ = dzdp f F G F G + F G F G FG * * p z z p V a a a a * ikz * ikz * i ( a ˆ a ˆ ) ikz * ikz * ( e ˆ e ˆ ) dzdp f 1+ p + aa e e e e A + A * iks + kvaa dz a e a e A,, F G F G Autonomization: FG = FG + ith H = H+ E a a t E E t Time dependent transformation (canonical): fˆ ( θ, p ) = f ( z, p ) ith θ = ( k + k ) z kt vanishing ikt aˆ = ae ˆ * E = E + Vk aa and tˆ = t Bracket: canonical Hamiltonian: H = ˆ i i k θ θ ˆˆ ( ˆ ˆ ) θ + + + * * d dp f 1 p aa ia ae a e a p k + k
Bonifacio s FEL model Resonance condition: p = p + p ith p p R eak radiated field: aˆ γ 1 + p R R, ( k k ) d dpfˆ δf δg δf δg i = + θ F G F G Bracket: FG + ˆ ˆ ˆ ˆ * * θ p p θ f kv aˆ aˆ aˆ δ aˆ Hamiltonia ˆ 1 + a p ia iθ * iθ n: H = dθdpf ( θ, p ) 3 ( aˆe aˆ e ) γ γ R R Normalization R ϕ i Transformation (canonical) into intensity/phase a = i Ie, δf δg δf δg Bracket: canonical FG = dθdpf F G F G + θ p p θ ϕ I I ϕ p Hamiltonian : H = dθdpf ( θ, p) + I dθdpf ( θ, p) cos( θ ϕ)
Outlook:on the use of reduced Hamiltonian models 1.5 H N N p N j I = + cos( θ ϕ) j N j= 1 j= 1 I/N 1 0.5 0 0 100 00 z Long-range interacting systems : QSS, transition to equilibrium, Gyrokinetics: understand plasma disruption, control, Contact: chandre@cpt.univ-mrs.fr References: Bachelard, Chandre, Vittot, PRE (008) Chandre, Brizard, in preparation.