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Hamiltonian formulation of reduced Vlasov-Maxwell equations
Cristel CHANDRE
Centre de Physique Théorique – CNRS, Marseille, France
Contact: [email protected]
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: [email protected]
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,…
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