[PPT] - Ham iltonian form ulation of reduced Vlasov-Maxw ell equations PowerPoint Presentation

SLIDE 1

Ham iltonian form ulation of reduced Vlasov-Maxw ell equations

Cristel CHANDRE Centre de Physique Théorique – CNRS, Marseille, France

Contact: chandre@cpt.univ-mrs.fr

SLIDE 2

Outline Outline

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 charged particles interacting with fields: plasma physics (tokamaks), free electron lasers Here: reduced models of such systems (easier simulation, better understanding of the dynamics)

SLIDE 3

Motion of a charged particle in electrom agnetic fields Motion of a charged particle in electrom agnetic fields

( ) ( ) ( ) { } { } { }

2

, , , , , 2 , , ⎛ ⎞ ⎟ ⎜ ⎟ − ⎜ ⎟ ⎜ ⎟ ⎜ ∂ ∂ ∂ ∂ ⎝ ⎠ = + = ⋅ − ⋅ ∂ ∂ ∂ ∂ ⎧ ⎪ ∂ ⎪ = − ⎪ ⎪ ∂ ⎪ ⎨ ⎪ ∂ ⎪ = ⎪ ⎪ ∂ = = ⎪ ⎩ p A q p q q q p p q q p q p p q

with

equations of motion : e t c f g f g H t eV H H H t f g m H

> > in canonical form > > in non-canonical form: physical variables

( ) ( ) { } { } { }

2 2

1 1 , , , , , , 2 ⎛ ⎞ ∂ ∂ ⎟ ⎜ ⎟ + ⋅ × ⎜ ⎟ ⎜ ⎟ ⎜∂ ∂ ⎝ ⎠ ⎛ ⎞ ∂ ∂ ∂ ∂ ⎟ ⎜ ⎟ = + = ⋅ − ⋅ ⎜ ⎟ ⎜ ⎟ ⎜∂ ∂ ∂ ∂ ⎝ ⎠ ⎧ ⎪ = ⎪ ⎪ ⎪ ⎛ ⎞ ⎨ × ⎟ ⎜ ⎪ ⎟ = + ⎜ ⎪ ⎟ ⎜ ⎟ = = ⎜ ⎪ ⎝ ⎠ ⎪ ⎩ B v x x v x v x x v v x v v v v B v E

with

equations of motion : f g f g H t m eV t f g m e m c e f H H g m c

( ) 1 , e t m c ⎛ ⎞ ⎟ ⎜ = − ⎟ ⎜ ⎟ ⎜ ⎝ ⎠ = v p A q x q

gyroscopic bracket

SLIDE 4

Definition: Ham iltonian system Definition: Ham iltonian system

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

{ }

a scalar function

, the Hamiltonian

a Poisson bracket

, with the properties antisymmetric , , Leibnitz law , , , Jacobi identity , , , , , ,

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

, dF F H dt F H = =

SLIDE 5

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

f t f t t t N = 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 f f t m c d f dt t f m d xd v f ⎛ ⎞ ∂ ∂ ⎟ ⎜ ⎟ = − ⋅ ∇ − + × ⋅ ⎜ ⎟ ⎜ ⎟ ⎜ ∂ ∂ ⎝ ⎠ ∂ ⎡ ⎤ ⎡ ⎤ = = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ∂ ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ +

∫∫

v v E v B v F F F F H H

3 3

with , , eV d xd v f f f ⎛ ⎞ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎝ ⎠ ⎧ ⎫ ⎪ ⎪ δ δ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ δ δ ⎪ ⎪ ⎩ = ⎥ ⎭ ⎡ ⎤ ⎢ ⎣ ⎦

∫∫

F G G F

SLIDE 6

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

( )

3 3 3 3 3 2 2

an example:

( , ) , ,

functional derivatives

= + 1

here:

and 2

therefore:

f e d v f d xd v f t f f f f d xd v O f e m eV f 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 H H F F F H

3

with f e d v f ∂ ⎡ ⎤ − ⋅ = ⎢ ⎥ ⎣ ⎦ ∂

∫

J J v x

SLIDE 7

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

4 4 f e f f t m c c t c t ⎛ ⎞ ∂ ∂ ⎟ ⎜ ⎟ = − ⋅ ∇ − + × ⋅ ⎜ ⎟ ⎜ ⎟ ⎜ ∂ ∂ ⎝ ⎠ ∂ = − ∇× ∂ ∂ = ∇× − π ∂ ∇ ⋅ = πρ ∇ ⋅ = v v E B v B E E B J E B where and

Vlasov Vlasov-

Maxw ell equations: self

Maxw ell equations: self-

consistent dynam ics

consistent dynam ics

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

SLIDE 8

3 2 2 3 3 3 3 3 2 3 3

2 , , , 4 4 , 8 m d xd v f d xd v f f e f d 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 B E E B B B E F G F G H F F G F G F G G Hamiltonian : with

3

, , , d f dt t d p f ∂ ⎡ ⎤ ⎡ ⎤ = = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ − ⎦ ∂

∫

E E B B F F F F H Equation of motion for : Remark: et are conserved quantities antisymmetry, Leibnitz, div div Jacobi

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

Vlasov Vlasov-

Maxw ell equations... still a Ham iltonian system

Maxw ell equations... still a Ham iltonian system

SLIDE 9

Elimination (or decoupling) of fast time and small spatial scales for a better

understanding 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 , 4 where ,

R R R R

c t c t ∇ ⋅ = πρ ∂ = ∇× − π ∂ = + π = − π ∂ ρ = ρ + ∇ ⋅ = − ∇× − ∂ D H D D H J D E P H B M P P J J M

From m icroscopic to m acroscopic Vlasov From m icroscopic to m acroscopic Vlasov-

Maxw ell equations

Maxw ell equations

SLIDE 10

Reduced Reduced fields fields as Lie as Lie transform s transform s of

f f

f, , E E and and B B

( ) ( ) ( )

Given a functional , , , , , , , we define some new fields as 1 , , , 2 1 e , , , 2 , t t f t F f f f

−

⎡ ⎤ ⎢ ⎥ ⎣ ⎦ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ − + + ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎛ ⎞ ⎛ ⎞ ⎣ ⎦ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎡ ⎤ ⎟ ⎟ ⎡ ⎤ ⎡ ⎤ ⎜ ⎜ = = − + + ⎟ ⎟ ⎢ ⎥ ⎢ ⎥ ⎜ ⎜ ⎢ ⎥ ⎟ ⎟ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎟ ⎟ ⎜ ⎜ ⎝ ⎠ ⎝ ⎠ ⎡ − ⎢ ⎣ E x B x x v E E E D E H B B B B

S

L

S S S S S S S S 1 , , 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 x

S

L

S S e , resulting in a new Hamiltonian and a new Poisson bracket...

−

=

S

L

F F

SLIDE 11

( ) ( )

3

1 e 1 4 1 1 e 4 4 so that 4 Reduced evolution operator e e e e ,e , e c d vf m f c t t

− − − −

⎛ ⎞ δ ∂ δ ⎟ ⎜ ⎟ = − = ∇× − + ⎜ ⎟ ⎜ ⎟ ⎜ π δ ∂ δ ⎝ ⎠ δ = − = ∇× + π δ ⎧ ⎪ = + π ⎪ ⎨ ⎪ = − π ⎪ ⎩ ⎛ ⎞ ∂ ∂ ⎟ ⎜ ⎟ ≡ ⎜ ⎟ ⎜ ⎟ ⎜ ∂ ∂ ⎝ ⎠ ⎡ ⎤ ⎡ ⎤ = = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦

∫

P E B v M B E D E P H B M

S

S S S S S S

L L L L L L L

S S S F F F H F H

Polarization Polarization, , m agnetization m agnetization, , reduced reduced density density , , etc etc… …

SLIDE 12

Reduced Reduced Vlasov Vlasov-

Maxw ell

Maxw ell equations equations

4 c t c t ∂ = ∇× − π ∂ ∂ = − ∇× ∂ D H J H D , 4 , 4 4

R

c t t c c t t t ∂ ∂ ⎡ ⎤ = + − = ∇× − π ⎢ ⎥ ⎣ ⎦ ∂ ∂ ∂ ∂ ∂ ⎡ ⎤ = + − = − ∇× − π + π ∇× ⎢ ⎥ ⎣ ⎦ ∂ ∂ ∂ D D D H J H H M H D P H H H H Reduced Vlasov equation 4 , F e F F t m c e f F f f f m ⎛ ⎞ ∂ ∂ ⎟ ⎜ ⎟ = − ⋅ ∇ − + × ⋅ ⎜ ⎟ ⎜ ⎟ ⎜ ∂ ∂ ⎝ ⎠ ⎧ ⎫ ⎪ ⎪ δ π ∂ δ ⎪ ⎪ = − − ⋅ + ⎨ ⎬ ⎪ ⎪ δ ∂ δ ⎪ ⎪ ⎩ ⎭ v v D H v v E

S

S

guiding center theory / gyrokinetics

SLIDE 13

What What S S ? ?

e F f

−

⎛ ⎞ ⎛ ⎞ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ = ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎟ ⎟ ⎜ ⎜ ⎝ ⎠ ⎝ ⎠ D E H B

S

L

Elimination of small spatial and fast time (averaging) scales of f(x,v,t) :

guiding-center gyrokinetics reduced 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 , f f f f ⎡ ⎤ = + δ δ + = ⎢ ⎥ ⎣ ⎦ P S

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

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

SLIDE 14

Strategies Strategies to to reduce reduce Vlasov Vlasov-

Maxw ell

Maxw ell equations equations

>rigorous: e > non-rigorous: truncate the Hamiltonian system

the equations of motion
the Hamiltonian and the Poisson bracket

> the canonical version provides a way

−

=

S

L

H H

ut...

SLIDE 15

Reduced Reduced m odel for the Free Electron Laser m odel for the Free Electron Laser

From : Vlasov-Maxwell Hamiltonian To: Bonifacio’s reduced FEL Hamiltonian model … in a Ham iltonian way

( ) ( ) ( )

2

, , , 2 sin 2 , p H f I d dp f p I I ⎛ ⎞ ⎟ ⎜ ⎡ ⎤ ⎟ ϕ = θ θ + θ −ϕ ⎜ ⎟ ⎢ ⎥ ⎜ ⎣ ⎦ ⎟ ⎜ ⎝ ⎠ ϕ

∫∫

with intensity and phase of the electromagnetic wave

( )

2 2 3 3 3 2

, 1 , 2 , d xd p f d x f ⎡ ⎤ = + ⎢ + ⎣ + ⎥ ⎦

∫∫ ∫

x p E p E B B H

SLIDE 16

A Free Electron A Free Electron… … w hat w hat? ?

rms

u LEL

K

2

2 (1

) 2 λ λ γ = +

SLIDE 17

Vlasov Vlasov-

Maxw ell: canonical version

Maxw ell: canonical version

( ) ( ) ( )

( )

3 3 3 3 3

, , ( , , ) , , 1 , d xd p f f d xd p f f f d f f f x 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 A

m

m m m m m m m

Bracke Change of va t: Hamiltonian: riables: canonical

F

G F G F G F F G G H

( ) ( ) ( ) ( ) ( )

2 2 3 2 3 2 2 3 3

2 , , 1 2

w w w

d x d x f 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 transformation racket: Hamiltonia n: )

H

( )

2 *

e 2 ˆ ˆ e 2

w w

ik z ik z w w w

a

−

= +

∫

A A e e helicoidal undulator:

SLIDE 18

One m ode for the One m ode for the radiated radiated field field

( ) ( )

* * * *

ˆ ˆ ˆ ˆ ˆ 2 2 ˆ ˆ 2

ikz ikz ikz ikz

i i a a k a a a x y

− −

+ = − − = = + x y A 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

(

)

2 3 3 2 * * * * * 2 * 3 3 *

ˆ e e 2 , ˆ ˆ 1 2 e e

ikz ikz w w ikz ik

d xd p f aa i a d xd p f f f f f ik V a a a a ikS k Vaa z a a a d

− −

⎛ ⎞ ∂ ∂ ∂ ∂ ⎟ ⎜ ⎟ + − ⎜ ⎟ ⎜ ⎟ ⎜ ∂ ∂ ∂ ∂ ⎝ ⎠ + ⎡ ⎤ δ ∂ δ ∂ δ δ ⎢ ⎥ ∇ ⋅ − ⋅ ∇ ⎢ ⎥ δ ∂ δ ∂ δ δ ⎢ ⎡ ⎤ − = ⎢ ⎥ ⎣ + + − ⎦ = ⋅ ⎥ − + ⎣ ⎦ −

∫ ∫ ∫∫ ∫

p e e A A e p p

m m m m m m

b Bracket: Hamiltonian: le) F G F G F G F G G H F

( )

*

ˆ

z w

⋅ ∇× e A

z

L: interaction length V: interaction volume

ˆ x

ˆ y

SLIDE 19

Dim ensional Dim ensional reduction reduction

( )

( ) (

) ( ) ( ) ( ) ( )

( ) (

) ( ) ( ) ( ) ( )

, , , , , 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 , ˆ 1 2 e e 2

ikz ikz w w

dzdp f p aa ik V a a a a ik dzdp f p f z S k Vaa f p f i z a f a

−

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

∫∫ ∫∫

e e A A

Bracket:

Hamilton

n at the cen

ian: ter) F G F F G F G F G G H

( ) ( ) ( )

( )

* 2 * *

, , ˆ , , ˆ e ˆ ˆ e e ˆ ˆ

ikz ikz w t w ik

E t 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

a

a

Autonomization: with Time dependent transformation (canonical): Bracke with and F G F G F G F G H H

(

)

2 * * 2

ˆ ˆˆ ˆ ˆ 1 e e

i i w w w

k d dp f p aa ia a a a p k k

θ − θ

= ⎛ ⎞ ⎟ ⎜ ⎟ ⎜ θ + + − − + − ⎟ ⎜ ⎟ ⎟ ⎜ + ⎝ ⎠

∫∫

can

t: Ha

n

mi ical ltonian: H

vanishing

SLIDE 20

Bonifacio Bonifacio’ ’s s FEL m odel FEL m odel

( )

* * 2

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 1 ˆ ,

w R R R R

k k d dp f p p f f f f p p p p p i kV a a a a p a ⎡ ⎤ ∂ δ ∂ δ ∂ δ ∂ δ ⎢ ⎥ + θ − ⎢ ⎥ ∂θ ∂ ∂ = + γ ≡ + ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ ∂θ δ δ ⎛ ⎞ ∂ ∂ ∂ δ ∂ ⎟ ⎜ ⎟ + − ⎜ ⎟ ⎜ δ ⎣ ⎟ ⎜ ∂ ∂ ∂ ∂ ⎠ ⎦ ⎝

∫∫

Resonance condition:

with weak radiated field: Bracket: Hamiltonia

F

G F G F G F G F G

( )

2 2 * 3

1 ˆ ˆ ˆ e e 2 , ,

i i w w R i R

a i d dp f f 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 G F G

( ) ( ) ( )

2

2 s , 2 , co I d dpf p I I f p d dp f p ∂ ∂ ∂ ∂ + − ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ = + θ θ θ − ∂ϕ ∂ θ ϕ ∂ ∂ θ ϕ

∫ ∫ ∫ ∫ ∫∫

Hamiltonian : H F G F G

SLIDE 21

Outlook:on Outlook:on the use of the use of reduced reduced Ham iltonian Ham iltonian m odels m odels

Contact: chandre@cpt.univ-mrs.fr References: Bachelard, Chandre, Vittot, PRE (2008) Chandre, Brizard, in preparation.

2 1 1

2 cos( ) 2

N N j j j j N

H N I p

= =