Hamiltonian formulation of reduced Vlasov-Maxwell equations

Cristel CHANDRE

Centre de Physique Théorique – CNRS, Marseille, France

Contact: chandre@cpt.univ-mrs.fr

OutlineOutline- Hamiltonian description of charges particles and electromagnetic fields

- Reduction of Vlasov-Maxwell 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)

importance of stability vs instability in devices involving a large number of chargedparticles interacting with fields: plasma physics (tokamaks), free electron lasers

Here: reduced models of such systems (easier simulation, better understanding of the dynamics)

Motion of a charged particle in electromagnetic fieldsMotion of a charged particle in electromagnetic fields

( )( )

( ) { }

{ }

{ }

2

,

, ,

,

,2

,

,

⎛ ⎞⎟⎜ ⎟−⎜ ⎟⎜ ⎟⎜ ∂ ∂ ∂ ∂⎝ ⎠= + = ⋅ − ⋅

∂ ∂ ∂ ∂⎧⎪ ∂⎪ = −⎪⎪ ∂⎪⎨⎪ ∂⎪ =⎪⎪ ∂

=

=⎪⎩

p A qp q q

q p p q

qp

qp

p

q

with

equations of motion :

e tc f g f gH t eV

H

HH

t f gm

H

>> in canonical form

>> in non-canonical form: physical variables

( ) ( ) { }

{ }{ }

2

21 1,

,

,

, , ,2

⎛ ⎞∂ ∂ ⎟⎜ ⎟+ ⋅ ×⎜ ⎟⎜ ⎟⎜∂ ∂⎝ ⎠

⎛ ⎞∂ ∂ ∂ ∂ ⎟⎜ ⎟= + = ⋅ − ⋅⎜ ⎟⎜ ⎟⎜∂ ∂ ∂ ∂⎝ ⎠⎧⎪ =⎪⎪⎪ ⎛ ⎞⎨ × ⎟⎜⎪ ⎟= +⎜⎪ ⎟⎜ ⎟

=

=⎜⎪ ⎝ ⎠⎪⎩

Bv

x x

v x v xx v v x

v

vv v

B

v

E

with

equations of motion :

f g f gH t m eV t f gm

em c

e f

H

H

gm c

( )1

,e t

m c⎛ ⎞⎟⎜= − ⎟⎜ ⎟⎜⎝ ⎠

=

v p A q

x q

gyroscopic bracket

Definition: Hamiltonian systemDefinition: Hamiltonian system

{ }{ } { }

{ } { } { }{ }{ } { }{ } { }{ }

- a scalar function , the Hamiltonian

- a Poisson bracket , with the properties

antisymmetric , ,

Leibnitz law , , ,

Jacobi identity , , , , , , 0

-

H

F G

F G G F

F GK F G K G F K

F G K K F G G K F

= −

= +

+ + =

{ }

{ }

equations of motion

,

- a conserved quantity

, 0

dFF H

dt

F H

=

=

Eulerian version: case of a density of charged particlesEulerian version: case of a density of charged particles

( )( ) ( )( ) ( )( )

- density of particles in phase space , ,

1 example: , , Klimontovitch distribution

- evolution given by the Vlasov equation

i ii

f t

f t t tN

= δ − δ −∑

x v

x v x x v v

3 3 2

- Eulerian, not Lagrangian:

for any observable , we have ,

- still a Hamiltonian system

Hamiltonian :2

f e ff

t m c

df

dt t

fm

d xd v f

⎛ ⎞∂ ∂⎟⎜ ⎟= − ⋅∇ − + × ⋅⎜ ⎟⎜ ⎟⎜∂ ∂⎝ ⎠

∂⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦∂

⎡ ⎤ =⎢ ⎥⎣ ⎦ +∫∫

vv E

v

Bv

F FF F H

H

3 3 with , ,

eV

d xd v ff f

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠

⎧ ⎫⎪ ⎪δ δ⎪ ⎪⎨ ⎬⎪ ⎪δ δ⎪ ⎪⎩=⎥

⎭⎡ ⎤⎢⎣ ⎦ ∫∫F

GG

F

Eulerian version: case of a density of charged particlesEulerian version: case of a density of charged particles

( )

( )

( )

30 0

3 3

3 3 2

20

- an example: ( , )

, ,

- functional derivatives

= +

1- here: and

2

- therefore:

f e d v f

d xd v ft f f

f f d xd v Of

e m eVf f

t

⎡ ⎤ρ =⎢ ⎥⎣ ⎦⎧ ⎫⎪ ⎪∂ρ δρ δ⎪ ⎪⎡ ⎤= ρ = ⎨ ⎬⎢ ⎥⎣ ⎦ ⎪ ⎪∂ δ δ⎪ ⎪⎩ ⎭

δ⎡ ⎤ ⎡ ⎤+ φ φ + φ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ δ

δρ δ= δ − = +

δ δ

∂ρ=

∂

∫

∫

∫

x x v

x x v

HH

FF F

H

3 with f e d v f∂ ⎡ ⎤− ⋅ =⎢ ⎥⎣ ⎦∂ ∫J J vx

Variables: particle density f(x,v,t), electric field E(x,t), magnetic field B(x,t)

4

4 0

f e ff

t m c

ct

ct

⎛ ⎞∂ ∂⎟⎜ ⎟= − ⋅∇ − + × ⋅⎜ ⎟⎜ ⎟⎜∂ ∂⎝ ⎠

∂= − ∇×

∂∂

= ∇× − π∂

∇⋅ = πρ ∇ ⋅ =

vv E B

v

BE

EB J

E Bwhere and

VlasovVlasov--Maxwell equations: selfMaxwell equations: self--consistent dynamicsconsistent dynamics

> description of the dynamics of a collisionless plasma (low density)

3

2 2

3

3 3

3

32

3

3

2, ,

,

4

4

,

8m

d xd v f

d xd v ff

e fd xd v

m f

d x

x

f

f

f

c d

⎧ ⎫⎪ ⎪δ δ⎪ ⎪⎨ ⎬⎪ ⎪δ δ⎪ ⎪

⎡ ⎤ = +⎢ ⎥⎣ ⎦

⎡ ⎤

+

π

⎡ ⎤δ δ δ δ⎢ ⎥+ π ⋅∇× −∇× ⋅⎢ ⎥δ δ δ δ

⎡ ⎤π ∂ δ δ δ δ⎢ ⎥+ ⋅ −⎢ ⎥∂ δ δ δ δ⎣ ⎦

⎩=⎢ ⎥⎣ ⎦

⎭

⎣ ⎦

∫

∫

∫

∫∫

∫∫

∫

v

v

E E

E BE

E B B

B

EF G F G

H

F

F G F G

F GG

Hamiltonian :

with

3

, , ,d

fdt t

d p f

∂⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣

−

⎦∂

∫

E

E

B

B

F FF F HEquation of motion for :

Remark: et are conserved quantities

antisymmetry, Leibnitz,

div div

Jacobi

Morrison, PLA (1980)Marsden, Weinstein, Physica D (1982)

VlasovVlasov--Maxwell equations... still a Hamiltonian systemMaxwell equations... still a Hamiltonian system

- Elimination (or decoupling) of fast time and small spatial scales for a betterunderstanding of complex plasma phenomena

reduced polarization density / magnetization current / polarization current density

- Can we represent the reduced Vlasov-Maxwell equations as a Hamiltonian system?Hint: use of Lie transforms

- Deliverables: Expressions of the polarization P and magnetization M vectors

- reduced Maxwell equations in terms of and

4

4

4 , 4where

,

R

R

R R

ct

ct

∇⋅ = πρ∂

= ∇× − π∂= + π = − π

∂ρ = ρ +∇⋅ = − ∇× −

∂

D HDD

H J

D E P H B MP

P J J M

From microscopic to macroscopic VlasovFrom microscopic to macroscopic Vlasov--Maxwell equationsMaxwell equations

ReducedReduced fieldsfields as Lie as Lie transformstransforms of of ff, , EE and and BB

( ) ( ) ( )Given a functional , , , , , , , we define some new fields as

1, , ,

21

e , , ,2

,

t t f t

F ff f

−

⎡ ⎤⎢ ⎥⎣ ⎦

⎡ ⎤⎡ ⎤ ⎡ ⎤− + +⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎛ ⎞ ⎛ ⎞ ⎣ ⎦⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜ ⎡ ⎤⎟ ⎟ ⎡ ⎤ ⎡ ⎤⎜ ⎜= = − + +⎟ ⎟ ⎢ ⎥ ⎢ ⎥⎜ ⎜ ⎢ ⎥⎟ ⎟ ⎣ ⎦ ⎣ ⎦⎣ ⎦⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠ ⎡− ⎢⎣

E x B x x v

E E ED EH B B B BSL

S

S S S

S S S

S1

, ,2

Remark: If the variable is only a function of

then e is only a function of

The functionals transforms into

f

−

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎡ ⎤⎤ ⎡ ⎤⎜ ⎟+ +⎜ ⎟⎥ ⎢ ⎥⎢ ⎥⎦ ⎣ ⎦⎣ ⎦ ⎟⎜⎝ ⎠

χ

χ

x

xSL

S S

e ,

resulting in a new Hamiltonian and a new Poisson bracket...

−= SLF F

( )

( )

31e 1

41

1 e4

4so that

4

Reduced evolution operator e e

e e , e ,

ec d vf

m f

c

t t

−

−

−

−

⎛ ⎞δ ∂ δ ⎟⎜ ⎟= − = ∇× − +⎜ ⎟⎜ ⎟⎜π δ ∂ δ⎝ ⎠δ

= − = ∇× +π δ

⎧⎪ = + π⎪⎨⎪ = − π⎪⎩

⎛ ⎞∂ ∂ ⎟⎜ ⎟≡ ⎜ ⎟⎜ ⎟⎜∂ ∂⎝ ⎠⎡ ⎤ ⎡ ⎤= = ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦

∫P EB v

M BE

D E PH B M

S

S

S S

S S S

L

L

L L

L L L

S S

S

FF

F H F H

PolarizationPolarization, , magnetizationmagnetization, , reducedreduced densitydensity, , etcetc……

ReducedReduced VlasovVlasov--Maxwell Maxwell equationsequations

4ct

ct

∂= ∇× − π

∂∂

= − ∇×∂

DH J

HD

, 4

, 4 4

Rc

t t

c ct t t

∂ ∂ ⎡ ⎤= + − = ∇× − π⎢ ⎥⎣ ⎦∂ ∂∂ ∂ ∂⎡ ⎤= + − = − ∇× − π + π ∇×⎢ ⎥⎣ ⎦∂ ∂ ∂

D DD H J

H H MH D P

H H

H H

Reduced Vlasov equation

4,

F e FF

t m ce f

F f ff m

⎛ ⎞∂ ∂⎟⎜ ⎟= − ⋅∇ − + × ⋅⎜ ⎟⎜ ⎟⎜∂ ∂⎝ ⎠⎧ ⎫⎪ ⎪δ π ∂ δ⎪ ⎪= − − ⋅ +⎨ ⎬⎪ ⎪δ ∂ δ⎪ ⎪⎩ ⎭

vv D H

v

v ES S

guiding center theory / gyrokinetics

WhatWhat SS ??

e

F f

−

⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜=⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠

D EH BSL

- Elimination of small spatial and fast time (averaging) scales of f(x,v,t) : guiding-centergyrokineticsreduced models for free electron lasers

-Use of KAM algorithms (at least one step process)

- Advantages: preserve the structure of the equations,invertible, symbolic calculus

( )For F , homological equation , 0f f f f⎡ ⎤= + δ δ + =⎢ ⎥⎣ ⎦P S

collisionless plasmas (low frequency phenomena)⎫⎪⎪⎬⎪⎪⎭

Brizard, Hahm, Rev. Mod. Phys. (2007)

StrategiesStrategies to to reducereduce VlasovVlasov--Maxwell Maxwell equationsequations

>rigorous: e

> non-rigorous: truncate the Hamiltonian system

- the equations of motion

- the Hamiltonian and the Poisson bracket

> the canonical version provides a way

−= SLH H

out...

ReducedReduced model for the Free Electron Lasermodel for the Free Electron Laser

From: Vlasov-Maxwell Hamiltonian

To: Bonifacio’s reduced FEL Hamiltonian model

… in a Hamiltonian way

( ) ( )

( )

2

, , , 2 sin2

,

pH f I d dp f p I

I

⎛ ⎞⎟⎜⎡ ⎤ ⎟ϕ = θ θ + θ−ϕ⎜ ⎟⎢ ⎥ ⎜⎣ ⎦ ⎟⎜⎝ ⎠ϕ

∫∫with intensity and phase of the electromagnetic wave

( )2 2

33 3 2, 1,2

, d xd p f d xf⎡ ⎤ = +⎢+

⎣ +⎥⎦ ∫∫ ∫x pE

pEB

BH

A Free ElectronA Free Electron…… whatwhat??

rms

uLEL K

2

2(1 )

2λ

λγ

= +

VlasovVlasov--Maxwell: canonical versionMaxwell: canonical version

( )( ) ( )( )

3 3

3 3

3

, , ( , , )

, ,

1

, d xd p ff

d xd p f

f fd

f

f

fx

f

f

⎡ ⎤δ δ δ δ⎢ ⎥+ ⋅ −

= +

= −= ∇×

⎡ ⎤ =⎢ ⎥⎡ ⎤δ ∂ δ ∂ δ δ⎢ ⎥∇ ⋅ − ⋅∇⎢ ⎥δ ∂ δ ∂ δ δ

⋅⎢ ⎥δ δ δ δ⎣⎢ ⎥

=

⎣ ⎦

+

⎦⎦

⎣ ∫

∫∫

∫∫

E B Y A

x p x p A x

E Y

A Y Y

B A

p p Amm m m

m

m

m

m

Bracke

Change of va

t:

Hamiltonian:

riables:

canonical

F GF G F GF

F GG

H ( )

( ) ( ) ( )

( )

2 2

3

2

3

2

23 3

2

, ,

12

w

w

w

d x

d xf

t

p

t

d xd

−+ ∇×

+ ∇× ⋅∇× + ∇

+

+

= − ++×

−∫∫

∫p A

p A A

Y A

Y A A

A A x A

A

x A x

m

B

Translation of by a constant function (external field-undulator):

canonical (canonical transformationracket:

Hamiltonia n:

)

H

( )

2

*e

2

ˆ ˆe2

w wik z ik zww w

a −= +

∫

A A e e helicoidal undulator:

One mode for the One mode for the radiatedradiated fieldfield

( )

( )

* *

* *

ˆ ˆˆ ˆ ˆ

2 2

ˆ ˆ2

ikz ikz

ikz ikz

i ia a

ka a

a x y

−

−

+= − − =

= +

x yA e e e

Y e e

Paraxial approximation and circularly polarized radiated field

e e where

e e

Remark: does not depend on and but depends on time (dynamical varia

( ) 23 3 2 * * *

* *

2 *

3 3

*ˆe e2

,

ˆ ˆ1 2 e eikz ikzw w

ikz ik

d xd p f aa i a

d xd p ff f f f

ikV a aa a

ikSk Vaa z a a

a

d

−

−

⎛ ⎞∂ ∂ ∂ ∂ ⎟⎜ ⎟+ −⎜ ⎟⎜ ⎟⎜ ∂ ∂∂ ∂⎝ ⎠

+

⎡ ⎤δ ∂ δ ∂ δ δ⎢ ⎥∇ ⋅ − ⋅∇⎢ ⎥δ ∂ δ ∂ δ δ⎢⎡ ⎤

−

=⎢ ⎥⎣

+ + −

⎦

= ⋅

⎥

− +

⎣ ⎦

−∫

∫

∫∫

∫

p e e A A

e

p pmm m m

m

m

b

Bracket:

Hamiltonian:

le)

F G F GF G

F G G

H

F

( )*ˆzw

⋅ ∇×e A

z

L: interaction lengthV: interaction volume

xy

DimensionalDimensional reductionreduction

( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )

, , 0 0

0 ,

, , 0 0 0

x y

f f p t

t x y

f f z p x y x t y t

⊥ ⊥

⊥

⊥

⇒

= δ = =

=

= δ δ δ = = = =

x p x p p

p

x p p

The fields do not depend on and no transverse velocity dispersion

if

If then no modification of the distribution

if (injecti

( ) 2

* *

2

*

*

2 * *ˆ

,

ˆ1 2 e e

2

ikz ikzw w

dzdp f p aa

ikV a aa a

ik

dzdp fp f z

Sk Vaa

f p f

i

z

a

f

a −

⎛ ⎞∂ ∂ ∂ ∂ ⎟⎜ ⎟+ −⎜ ⎟⎜ ⎟⎜ ∂ ∂∂ ∂⎝ ⎠

⎡ ⎤∂ δ ∂ δ ∂ δ ∂ δ⎢ ⎥−⎢ ⎥∂ δ ∂ δ ∂

+ + − − ⋅ +

⎡ ⎤ =⎢ ⎥⎣ δ ∂ δ⎢

−

⎦

=

+

⎥⎣ ⎦

∫∫

∫∫

e e A A

Bracket:

Hamilton

on at the cen

ian:

ter)

F G F FG F GF G

G

H

( )

( ) ( ) ( )

*

2

*

*

, ,

ˆ , ,

ˆ eˆ

ˆe e

ˆ

ˆikz ikz

wt

w

ik

Et E E t

f p f z p k k z kt

a a

E E Vk aa t

z

t

d a a −

∂ ∂ ∂ ∂⎡ ⎤ ⎡ ⎤= + − = +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ∂

− ⋅∇×

∂ ∂ ∂

θ = θ = + −

=

= + =

∫ e e A

aa Autonomization: with

Time dependent transformation (canonical):

Bracke

with

and

F G F GF G F G H H

( )2 * * 2ˆ ˆˆ ˆ ˆ1 e ei iw w

w

kd dp f p aa ia a a a p

k kθ − θ=

⎛ ⎞⎟⎜ ⎟⎜θ + + − − + − ⎟⎜ ⎟⎟⎜ +⎝ ⎠∫∫

cant:

Ha

on

mi

ical

ltonian: H

vanishing

BonifacioBonifacio’’ss FEL modelFEL model

( ) * *

2

ˆˆ

ˆ

ˆ ˆˆ ˆˆ ˆ

1

ˆ,w

R R

R R

k k d dp fp pf f f f

p p p p p

ikV a aa

a p

a

⎡ ⎤∂ δ ∂ δ ∂ δ ∂ δ⎢ ⎥+ θ −⎢ ⎥∂θ ∂ ∂

= +

γ ≡ +

⎡ ⎤ =⎢ ⎥⎣ ⎦ ∂θδ δ

⎛ ⎞∂ ∂ ∂δ

∂ ⎟⎜ ⎟+ −⎜ ⎟⎜δ⎣ ⎟⎜ ∂ ∂∂ ∂ ⎠⎦ ⎝∫∫

Resonance condition: with

weak radiated field:

Bracket:

Hamiltonia

F G F GF G F GF G

( ) ( )2 2

*

3

1ˆ ˆ ˆe e2

,

, i iw w

R

i

R

a i

d dp ff p

a

a

p

f

d dp f p a a

i I

p f

θ −

− ϕ

θ⎛ ⎞+ ⎟⎜ ⎟⎜θ θ − −=

=

⎡ ⎤ =∂ δ ∂ δ ∂ δ ∂ δ

θ −∂θ δ ∂ δ ∂ δ ∂

⎟

⎢ ⎥⎣

⎜

θ

⎟⎜ γ ⎟⎜ γ

δ⎦

⎝ ⎠∫∫

Normalization

Transformation (canonical) into intensity/phase e

canonic

n:

Bracket al :F G

H

F GF G

( ) ( ) ( )2

2 s,2

, coI d dpf p

I Ifp

d dp f p

∂ ∂ ∂ ∂+ −

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

= + θ θ θ −

∂ϕ ∂

θ ϕ

∂ ∂

θ

ϕ

∫

∫

∫

∫

∫∫Hamiltonian : H

F G F G

Outlook:onOutlook:on the use of the use of reducedreduced HamiltonianHamiltonian modelsmodels

Contact: chandre@cpt.univ-mrs.fr

References: Bachelard, Chandre, Vittot, PRE (2008)Chandre, Brizard, in preparation.

2

1 1

2 cos( )2

N N

jj j

jNH

NIp

= =

ϕ= + −θ∑ ∑

0 100 2000

0.5

1

1.5

z

I/N

Long-range interacting systems : QSS, transition to equilibrium,…

Gyrokinetics: understand plasma disruption, control,…