Evolution & Actions · 2008. 3. 30. · The Darwin approximation “neglects retardation” So drop...
Transcript of Evolution & Actions · 2008. 3. 30. · The Darwin approximation “neglects retardation” So drop...
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Evolution & ActionsUnifying Approaches to the Darwin Approximation
Todd B. Krause, [email protected]
Institute for Fusion StudiesThe University of Texas at Austin
April 3, 2008Tech-X Corporation
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Collaborators
Amit S. ApteCentre for AppliedMathematics, Tata Instituteof Fundamental Research,Bangalore, [email protected]
Philip J. MorrisonInstitute for Fusion Studies,The University of Texas atAustin, Austin, [email protected]
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Outline
1 Approaches to the Darwin ApproximationQuasistatic ApproachOperator Approximation Approach
2 Equivalence of Darwin FormulationsDarwin IntegralEquivalent Potentials
3 Darwin Action PrinciplesDarwin Field ActionDarwin Particle-Field & Particle ActionsVlasov-Darwin Action
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Quasistatic ApproachOperator Approximation Approach
The Character of Physical Law
Ever since there have been laws. . .Thou shalt not kill (Exodus 20:13)If a man seizes a woman on a mountain, it is the man’s sin,and he is to die (Hittite Laws)G = 8πT (Einstein)
. . . there have been approximations to those lawsThou shalt not suffer a witch to live (Ex. 22:18)If [a man seizes a woman] in her house, it is the woman’ssin, and she is to die (Hitt. Laws)F = −GMmr/r2 (Newton)
C. G. Darwin (1920)
The order-(v/c)2 approximation to relativistic interaction ofclassical charged particles
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Quasistatic ApproachOperator Approximation Approach
The Character of Physical Law
Ever since there have been laws. . .Thou shalt not kill (Exodus 20:13)If a man seizes a woman on a mountain, it is the man’s sin,and he is to die (Hittite Laws)G = 8πT (Einstein)
. . . there have been approximations to those lawsThou shalt not suffer a witch to live (Ex. 22:18)If [a man seizes a woman] in her house, it is the woman’ssin, and she is to die (Hitt. Laws)F = −GMmr/r2 (Newton)
C. G. Darwin (1920)
The order-(v/c)2 approximation to relativistic interaction ofclassical charged particles
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Quasistatic ApproachOperator Approximation Approach
The Character of Physical Law
Ever since there have been laws. . .Thou shalt not kill (Exodus 20:13)If a man seizes a woman on a mountain, it is the man’s sin,and he is to die (Hittite Laws)G = 8πT (Einstein)
. . . there have been approximations to those lawsThou shalt not suffer a witch to live (Ex. 22:18)If [a man seizes a woman] in her house, it is the woman’ssin, and she is to die (Hitt. Laws)F = −GMmr/r2 (Newton)
C. G. Darwin (1920)
The order-(v/c)2 approximation to relativistic interaction ofclassical charged particles
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Quasistatic ApproachOperator Approximation Approach
Basics of the Darwin Approximation
The order-(v/c)2 approximation to the relativisticinteraction of classical charged particles:
L =n∑
a=1
(maq2
a2
+maq4
a8c2
)− 1
2
∑a 6=b
eaeb
rab
+12
∑a 6=b
eaeb
2c2rab[qa · qb + (qa · rab)(qb · rab)]
The interaction stems from the coupling with the fields ofother particles via Aµ = (φ, A), where
φ(x, t) =∑
b
eb
‖x− qb‖, A(x, t) =
∑b
eb[qb + (qb · rb)rb]
2c‖x− qb‖
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Quasistatic ApproachOperator Approximation Approach
So What Good Is It?
Grandson of the CharlesDarwinEvidently a ferventeugenist!
But that’s another talk. . .
Darwin’s originalmotivation
Bohr-Sommerfeld atom,hydrogen spectrum(Darwin 1920)
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Quasistatic ApproachOperator Approximation Approach
Uses in Plasma Physics
Mirrortron simulationsPlasma column withstationary ions,counterstreamingelectronsShear Alfven waves
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Quasistatic ApproachOperator Approximation Approach
Outline
1 Approaches to the Darwin ApproximationQuasistatic ApproachOperator Approximation Approach
2 Equivalence of Darwin FormulationsDarwin IntegralEquivalent Potentials
3 Darwin Action PrinciplesDarwin Field ActionDarwin Particle-Field & Particle ActionsVlasov-Darwin Action
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Quasistatic ApproachOperator Approximation Approach
Coulomb Gauge
We start with Maxwell’s equations for φ and A
∇2φ +1c
∂
∂t(∇ · A) = −4πρ,
∇2A− 1c2
∂2A∂t2 −∇
(∇ · A +
1c
∂φ
∂t
)= −4π
cJ
Employ Coulomb condition ∇ · A = 0, yielding
∇2φ = −4πρ,
∇2A− 1c2
∂2A∂t2 = −4π
cJ +
1c
∂∇φ
∂t
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Quasistatic ApproachOperator Approximation Approach
Coulomb Solution
The solution for the scalar potential is the friendly
φC(x, t) =
∫d3x ′
ρ(x′, t)r
The solution for the vector potential is more monstrous
AC(x, t) =1c
∫d3x ′
1r[J(x′, t ′)− r
(r · J(x′, t ′)
)]ret
+ c∫
d3x ′1r
∫ r/c
0dττ
[3r(r · J(x′, t − τ)
)−J(x′, t − τ)
]where r := ‖x− x′‖ and r = (x− x′)/r , “ret” denotes thatthe quantities in brackets are evaluated at the retardedtime t ′ = t − r/c
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Quasistatic ApproachOperator Approximation Approach
Coulomb Solution
The solution for the scalar potential is the friendly
φC(x, t) =
∫d3x ′
ρ(x′, t)r
The solution for the vector potential is more monstrous
AC(x, t) =1c
∫d3x ′
1r[J(x′, t ′)− r
(r · J(x′, t ′)
)]ret
+ c∫
d3x ′1r
∫ r/c
0dττ
[3r(r · J(x′, t − τ)
)−J(x′, t − τ)
]where r := ‖x− x′‖ and r = (x− x′)/r , “ret” denotes thatthe quantities in brackets are evaluated at the retardedtime t ′ = t − r/c
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Quasistatic ApproachOperator Approximation Approach
Quasistatic Coulomb Solution
The instantaneous, or ‘quasistatic’, form of the vectorpotential
Substitute J(x′, t − τ) → J(x′, t) in the second integralRemove “ret” from the first integral, i.e. evaluate the currentat the time t
AqsC (x, t) =
12c
∫d3x ′
1r[J(x′, t) + r
(r · J(x′, t)
)]Clearly the continuum analogue of
A(x, t) =∑
b
eb[qb + (qb · rb)rb]
2c‖x− qb‖
in Darwin’s original particle action
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Quasistatic ApproachOperator Approximation Approach
Turning the TablesWhat Equations Do φC & Aqs
C Satisfy?
Clearly φC satisfiesPoisson’s equation
∇2φC = −4πρ
What about AqsC ? Compute:
∇2AqsC = −4π
cJ +
1c
∫d3x ′
1r3 [J′ − 3(J′ · r)r]
What’s that mess on the right?
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Quasistatic ApproachOperator Approximation Approach
Turning the TablesVector Decomposition
We can decompose any vector into longitudinal andtransverse components
In particular, decompose the current
JT (x, t) =1
4π∇×∇×
∫d3x ′ J(x′, t)
r,
JL(x, t) = − 14π∇∫
d3x ′∇′ · J(x′, t)r
Re-express the longitudinal component
JL(x, t) =1
4π
∫d3x ′ 1
r3 [J′ − 3(J′ · r)r]
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Quasistatic ApproachOperator Approximation Approach
Turning the TablesEquation for Aqs
C
Just What We Need
∇2AqsC (x, t) = −4π
cJT (x, t)
This is Poisson’s equation in vector formThe source is not the full current, but only the transversecurrent JT
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Quasistatic ApproachOperator Approximation Approach
Outline
1 Approaches to the Darwin ApproximationQuasistatic ApproachOperator Approximation Approach
2 Equivalence of Darwin FormulationsDarwin IntegralEquivalent Potentials
3 Darwin Action PrinciplesDarwin Field ActionDarwin Particle-Field & Particle ActionsVlasov-Darwin Action
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Quasistatic ApproachOperator Approximation Approach
Do Over
Rather than approximate the solution, approximate thegoverning equations directlyArgue the form of the differential operator
The term (1/c2)∂2A/∂t2 leads to field retardationThe Darwin approximation “neglects retardation”So drop the time derivative of vector potential
Maxwell’s equations take the approximate form
∇2φ +1c
∂
∂t(∇ · A) = −4πρ,
∇2A−∇(∇ · A +
1c
∂φ
∂t
)= −4π
cJ
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Quasistatic ApproachOperator Approximation Approach
You Don’t Appreciate What You Have until. . .Continuity Equation
Now you’ve done it! You just lostconservation!
You can’t drop terms fromMaxwell’s equations withoutlosing the continuity equation
We now have the constraint
∂ρ
∂t+∇ · J = − 1
4πc∂2
∂t2∇ · A
For sources that conserve charge, we have a naturalansatz: ∇ · A = 0
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Quasistatic ApproachOperator Approximation Approach
AnSatz is Good Satz
We make the ansatz∇ · A = 0, yielding
∇2φ = −4πρ,
∇2A− 1c
∂∇φ
∂t= −4π
cJ
and see if the solution AOAsatisfies ∇ · AOA = 0
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Quasistatic ApproachOperator Approximation Approach
Problem Solved!Almost. . .
The first equation gives the Coulomb scalar potential, andthe second equation gives
AOA(x, t) =1c
∫d3x ′
J(x′, t)r
− 14πc
∂
∂t
∫d3x ′
∇′φC(x′, t)r
Check the ansatz
∇ · AOA =1c
∫d3x ′
1r
[∇′ · J(x′, t) +
∂ρ(x′, t)∂t
]
AOA is divergence-free when (ρ, J) satisfy chargeconservationConsistent with the ansatz!
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Quasistatic ApproachOperator Approximation Approach
Making It Work for YouEmploying Charge Conservation
Given that Coulomb gauge and charge conservation areequivalent, we can now use that to our advantage
Use the continuity equation in JL
JL(x, t) = − 14π∇∫
d3x ′∇′ · J(x′, t)r
=1
4π
∂∇φC(x, t)∂t
Then AOA satisfies
∇2AOA = −4π
c(J− JL) = −4π
cJT
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Quasistatic ApproachOperator Approximation Approach
Speak My Language!In Terms of Fields
Let E = ET + EL, with ∇ · ET = 0 and ∇× EL = 0. Thenthe preceding is equivalent to
1c
∂EL
∂t−∇× B = −4π
cJ, ∇ · EL = 4πρ,
1c
∂B∂t
+∇× ET = 0, ∇ · B = 0
for EL = −∇φ, ET = −(1/c)∂A/∂t , and B = ∇× AChanging gauge by function χ
4π
cJ = ∇× B′ +
1c
∂∇φ′
∂t= ∇× B +
1c
∂∇φ
∂t+
1c2
∂2χ
∂t2
so gauge invariant only when the function χ is of the sameorder in 1/c as A
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Quasistatic ApproachOperator Approximation Approach
Pros & Cons
ProsEliminates high-frequency, transverse electromagneticeffects
1c2
∂2A∂t2 ∼ 0 ⇐⇒ field time-variation
speed of light∼ 0
Retains electrostatic and low-frequency inductive effects
ConsNumerical instabilitiesDifficulties with boundaryconditions
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Quasistatic ApproachOperator Approximation Approach
Pros & Cons
ProsEliminates high-frequency, transverse electromagneticeffects
1c2
∂2A∂t2 ∼ 0 ⇐⇒ field time-variation
speed of light∼ 0
Retains electrostatic and low-frequency inductive effects
ConsNumerical instabilitiesDifficulties with boundaryconditions
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Quasistatic ApproachOperator Approximation Approach
Loch LommondThe High Road vs. the Low Road
L[A] = J
AC = L−1[J] L[A] = J
AqsC = L−1[J] AOA = L−1[J]
������
����
����
����
solve
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????
????
????
??
approx.
��� �� �� �� �� ��
approx.
��� �� �� �� �� ��
solve
oo //?=
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Quasistatic ApproachOperator Approximation Approach
Loch LommondToy Model
dxdt
=1
1− x
t =
∫ x1
(1− x)dx t =
∫ x2 dx1 + x
x1(t) = 1−√
1− 2t ≈ t x2(t) = et − 1
zztttttttttttsolve
$$JJJJJJJJJJJapprox.
��
approx.
��
solve
oo //?=
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Darwin IntegralEquivalent Potentials
Outline
1 Approaches to the Darwin ApproximationQuasistatic ApproachOperator Approximation Approach
2 Equivalence of Darwin FormulationsDarwin IntegralEquivalent Potentials
3 Darwin Action PrinciplesDarwin Field ActionDarwin Particle-Field & Particle ActionsVlasov-Darwin Action
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Darwin IntegralEquivalent Potentials
The Heart of the Matter
We have the Darwin field equations
∇2φOA(x, t) = −4πρ(x, t),
∇2AOA(x, t) = −4π
cJT (x, t)
A direct solution for the vector potential is
AOA(x, t) =1
4πc
∫d3x ′
r
[∇′ ×∇′ ×
∫d3x ′′
J(x′′, t)rx ′x ′′
]But Darwin’s original potential is
AqsC (x, t) =
12c
∫d3x ′
1r[J(x′, t) + r
(r · J(x′, t)
)]Are they equal?
If not, then AOA isn’t really Darwin
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Darwin IntegralEquivalent Potentials
Setting Up
Using vector identities, we bring AOA to the form
AOA =1c
∫d3x ′
J(x′, t)‖x− x′‖
+1
4πc
∫d3x ′
(J′ · ∇
)∇∫
d3x ′′
‖x− x′′‖‖x′ − x′′‖
The first term has the form we need for AqsC
Everything hinges on the second termWe need to perform the integral
ID :=
∫d3x ′′
‖x′′ − x‖‖x′′ − x′‖=
∫d3y
‖y‖‖y− z‖
letting y := x′′ − x′ and z := x− x′
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Darwin IntegralEquivalent Potentials
The Darwin IntegralWhat’s a Little Infinity among Friends?
The integral is simple if we use Legendre polynomials
1‖y− z‖
=∞∑
l=0
r l<
r l+1>
Pl(cos γ),
∫ 1
−1Pk (ξ)Pl(ξ)dξ =
2δkl
2l + 1
Then we just calculate
ID =
∫d3y
‖y‖‖y− z‖= 2π
∞∑l=0
∫ ∞
0
r l<
r l+1>
ydy∫ 1
−1P0(ξ)Pl(ξ)dξ
= 4π
∫ ∞
0
yr>
dy = 4π
∫ z
0
yz
dy + 4π
∫ ∞
z
yy
dy
= 4π · ∞ − 2πz
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Darwin IntegralEquivalent Potentials
Outline
1 Approaches to the Darwin ApproximationQuasistatic ApproachOperator Approximation Approach
2 Equivalence of Darwin FormulationsDarwin IntegralEquivalent Potentials
3 Darwin Action PrinciplesDarwin Field ActionDarwin Particle-Field & Particle ActionsVlasov-Darwin Action
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Darwin IntegralEquivalent Potentials
Regularize
The infinity is actually hidden behind a derivativeWe are free to subtract a term canceling the x′′ divergence
AOA =1c
∫d3x ′
J(x′, t)‖x− x′‖
+1
4πc
∫d3x ′
(J′ · ∇
)∇[∫
d3x ′′
‖x− x′′‖‖x′ − x′′‖
−∫
d3x ′′
‖x′′‖2
]=
1c
∫d3x ′
J(x′, t)‖x− x′‖
− 12c
∫d3x ′
(J′ · ∇
)∇‖x− x′‖
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Darwin IntegralEquivalent Potentials
There Can Be Only One!
This gives us exactly what we need
AOA =12c
∫d3x ′
J(x′, t) + r′ (r′ · J(x′, t))‖x− x′‖
= AqsC
The two potentials are in fact equal!
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Darwin Field ActionDarwin Particle-Field & Particle ActionsVlasov-Darwin Action
Outline
1 Approaches to the Darwin ApproximationQuasistatic ApproachOperator Approximation Approach
2 Equivalence of Darwin FormulationsDarwin IntegralEquivalent Potentials
3 Darwin Action PrinciplesDarwin Field ActionDarwin Particle-Field & Particle ActionsVlasov-Darwin Action
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Darwin Field ActionDarwin Particle-Field & Particle ActionsVlasov-Darwin Action
Got Milk?
Got Action?
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Darwin Field ActionDarwin Particle-Field & Particle ActionsVlasov-Darwin Action
Wanna Li’l Action. . .
Joe Monaghan (2004), talk entitled “SPH and Simulation”
Conservation of general properties may be more importantthan high order integration
Continuing. . .Lagrangians, when they can be used,are good because the physics can beadded consistently and invariants canbe satisfied more easily
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Darwin Field ActionDarwin Particle-Field & Particle ActionsVlasov-Darwin Action
Wanna Li’l Action. . .
Joe Monaghan (2004), talk entitled “SPH and Simulation”
Conservation of general properties may be more importantthan high order integration
Continuing. . .Lagrangians, when they can be used,are good because the physics can beadded consistently and invariants canbe satisfied more easily
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Darwin Field ActionDarwin Particle-Field & Particle ActionsVlasov-Darwin Action
Putting Words into ActionMaxwell Field Action
The standard Maxwell field action is
SMf [φ, A] =
∫dtd3x
{−ρφ +
1c
J · A +1
8π
[E2 − B2
]}where E = −∇φ− (1/c)∂A/∂t and B = ∇× A are justshorthand notationVariation gives Maxwell’s equations
∇2φ +1c
∂
∂t(∇ · A) = −4πρ,
∇2A− 1c2
∂2A∂t2 −∇
(∇ · A +
1c
∂φ
∂t
)= −4π
cJ
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Darwin Field ActionDarwin Particle-Field & Particle ActionsVlasov-Darwin Action
Putting Words into ActionDarwin Field Action
Similarly, we write a Darwin field action
SDf [φ, A] =
∫dtd3x
{−ρφ +
1c
J · A +1
8π
[E2 − E2
T − B2]}
where EL = −∇φ and ET = −(1/c)∂A/∂tVariation gives the equations
∇2φ +1c
∂
∂t(∇ · A) = −4πρ,
∇2A−∇(∇ · A +
1c
∂φ
∂t
)= −4π
cJ
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Darwin Field ActionDarwin Particle-Field & Particle ActionsVlasov-Darwin Action
Putting Words into ActionDarwin Field Action with Constraint
Setting ∇ · A = 0 is equivalent to imposing chargeconservation, and so
∇2φ = −4πρ,
∇2A− 1c
∂∇φ
∂t= −4π
cJ
=⇒
∇2φ = −4πρ,
∇2A = −4π
cJT
So we have an action for theDarwin field equations!
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Darwin Field ActionDarwin Particle-Field & Particle ActionsVlasov-Darwin Action
Outline
1 Approaches to the Darwin ApproximationQuasistatic ApproachOperator Approximation Approach
2 Equivalence of Darwin FormulationsDarwin IntegralEquivalent Potentials
3 Darwin Action PrinciplesDarwin Field ActionDarwin Particle-Field & Particle ActionsVlasov-Darwin Action
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Darwin Field ActionDarwin Particle-Field & Particle ActionsVlasov-Darwin Action
Brick by BrickDarwin Particle-Field Action
We may add the 2nd-order relativistic particle action,together with a coupling term
SD[q;φ, A]
=
∫dt
{∑a
[maq2
a2
+maq4
a8c2
]+∑
a
ea
∫d3xδ(3)(x− qa)
[−φ(x, t) +
qa
c· A(x, t)
]+
18π
∫d3x
[E2(x, t)− E2
T (x, t)− B2(x, t)]}
Variation over φ and A gives Darwin field equationsHow do we get particle equations?
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Darwin Field ActionDarwin Particle-Field & Particle ActionsVlasov-Darwin Action
Particle Equations
Vary φ and A to obtain the Darwin field equations
∇2φ = −4πρ, ∇2A = −4π
cJT
Express sources in terms ofparticles themselves
ρ(x, t) =∑
b
ebδ(3)(x− qb(t)),
J(x, t) =∑
b
ebqb(t)δ(3)(x− qb(t))
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Darwin Field ActionDarwin Particle-Field & Particle ActionsVlasov-Darwin Action
ParticlizationIt’s All the Rage!
Solve for potentials in terms of sources
φ(x, t) =
∫d3x ′K (x|x′)ρ(x′, t) =
∑b
ebK (x|qb),
Ai(x, t) =1c
∫d3x ′Kij(x|x′)Jj(x′, t) =
1c
∑b
ebKij(x|qb)qbj
where
K (x|x′) :=1
‖x− x′‖,
Kij(x|x′) :=1
2‖x− x′‖
[δij +
(xi − x ′i )(xj − x ′j )
‖x− x′‖2
]
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Darwin Field ActionDarwin Particle-Field & Particle ActionsVlasov-Darwin Action
On the MoveParticle Equations of Motion
Insert these solutions back into the action
SD[q] =
∫dt
{∑a
(maq2
a2
+maq4
a8c2
)
+12
∑a 6=b
eaeb
[K (qa|qb) +
qai qbj
c2 Kij(qa|qb)
]Vary over particle coordinates q
mqi = eEi +ec
(q× B)i − eq2
c2
(δij
2+
qi qj
q2
)ELj
E and B are shorthand for combinations of φ and Aφ and A are shorthand for combinations of q and q
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Darwin Field ActionDarwin Particle-Field & Particle ActionsVlasov-Darwin Action
Outline
1 Approaches to the Darwin ApproximationQuasistatic ApproachOperator Approximation Approach
2 Equivalence of Darwin FormulationsDarwin IntegralEquivalent Potentials
3 Darwin Action PrinciplesDarwin Field ActionDarwin Particle-Field & Particle ActionsVlasov-Darwin Action
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Darwin Field ActionDarwin Particle-Field & Particle ActionsVlasov-Darwin Action
Changing Viewpoints
We can Legendre (convex) transform the DarwinLagrangian to obtain the Darwin Hamiltonian
HD(q, p, t) =1
2m
(p− e
cA)2− 1
8m3c2
(p− e
cA)4
+eφ(q, t)
Letting P := p− (e/c)A(q, t), Hamilton’s equationsbecome
q =Pm
(1− P2
2m2c2
),
P = e[E +
(1− P2
2m2c2
)P
mc× B
]
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Darwin Field ActionDarwin Particle-Field & Particle ActionsVlasov-Darwin Action
Heretical RevelationsNoncanonical Hamiltonian Systems
Given a Hamiltonian H, and Z := (q, P), we may look atphase space actions of the form
S[Z ] =
∫ t1
t0
[θµ(Z , t)Zµ − H(Z , t)
]dt
Variation givesωµν Z ν = H,µ + ∂tθµ
with ωµν := θν,µ − θµ,ν
If Jνσωσµ = δνµ, then
Z ν = Jνµ (H,µ + ∂tθµ)
give Hamilton’s equations of motion
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Darwin Field ActionDarwin Particle-Field & Particle ActionsVlasov-Darwin Action
Darwin Phase Space Action
We form the noncanonical Darwin phase space action
SD[q, P] =
∫dt[(
P +ec
A)· q− P2
2m+
P4
8m3c2 − eφ
]The equations Z ν = Jνµ (H,µ + ∂tθµ) become(
qP
)=
(03×3 13×3−13×3
ec Bij
)( e ∂φ∂q + e
c∂A∂t(
1− P2
2m2c2
)Pm
)
where Bij := εijkBk
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Darwin Field ActionDarwin Particle-Field & Particle ActionsVlasov-Darwin Action
What’s in a Label?Passing to the Continuum
Envision a continuum of particles in phase spaceLabel each particle by the initial conditions Z0 := (q0, P0) ofits trajectory: q(q0, P0, t) and P(q0, P0, t), succinctly Z (Z0, t)This is an invertible map of phase space: Z0(Z , t)
Associated with each trajectory is a phase space numberdensity (distribution function) f0(Z0)
Like the initial conditions, f0(Z0) constant on the trajectoryTo find f at an observation point Ξ := (x,Π) at time t , weneed the f associated with the trajectory Z that hits Ξ at t
So start at Ξ and map back to initial conditions Z0:Ξ = Z (Z0, t). This gives the Euler-Lagrange map
f (Ξ, t)d6Ξ = f0(Z0(Ξ, t))d6Z0
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Darwin Field ActionDarwin Particle-Field & Particle ActionsVlasov-Darwin Action
The Whole ShebangVlasov-Darwin Action
Phase Space, Vlasov-Darwin Action
SVD[q, P;φ, A]
=
∫dt{∫
d6Z0 fR0(Z0)
[P · q− P2
2m+
P4
8m3c2
]+
∫d6Z0 fR0(Z0)
[−eφ(q, t) +
ec
A(q, t) · q]
+1
8π
∫d3x
[E2(x, t)− E2
T (x, t)− B2(x, t)]}
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Darwin Field ActionDarwin Particle-Field & Particle ActionsVlasov-Darwin Action
The Whole ShebangVlasov-Darwin Equations
Variation of SVD with respect to q and P yields the Darwinequations of motion
q =Pm
(1− P2
2m2c2
),
P = e[E +
(1− P2
2m2c2
)P
mc× B
]These together with the Euler-Lagrange map imply theVlasov equation in the Darwin approximation
∂fD∂t
+
(1− Π2
2m2c2
)Π
m· ∂fD
∂x
+ e[E +
(1− Π2
2m2c2
)Π
mc× B
]· ∂fD∂Π
= 0
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Summary
Darwin ActionsOne action encompasses both particle and field formulations ofthe Darwin approximation
Double troubleTwo common formulations of the Darwin approximation:particle and field
There Can Be Only One!The Darwin particle-field action encapsulates bothformulations from one consistent viewpointProvides a consistent check on orders of approximation
Vlasov-DarwinNoncanonical Hamiltonian systemsUnified action principle for Vlasov equation with Darwinparticle-field dynamics
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Summary
Darwin ActionsOne action encompasses both particle and field formulations ofthe Darwin approximation
Double troubleTwo common formulations of the Darwin approximation:particle and field
There Can Be Only One!The Darwin particle-field action encapsulates bothformulations from one consistent viewpointProvides a consistent check on orders of approximation
Vlasov-DarwinNoncanonical Hamiltonian systemsUnified action principle for Vlasov equation with Darwinparticle-field dynamics
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Summary
Darwin ActionsOne action encompasses both particle and field formulations ofthe Darwin approximation
Double troubleTwo common formulations of the Darwin approximation:particle and field
There Can Be Only One!The Darwin particle-field action encapsulates bothformulations from one consistent viewpointProvides a consistent check on orders of approximation
Vlasov-DarwinNoncanonical Hamiltonian systemsUnified action principle for Vlasov equation with Darwinparticle-field dynamics
Todd B. Krause Unifying Darwin
Approaches to the Darwin ApproximationEquivalence of Darwin Formulations
Darwin Action PrinciplesSummary
Select Bibliography
C. G. DarwinThe Dynamical Motions of Charged Particles.Rev. Mod. Phys., 39(233):537–551, 1920.
A. Kaufman and P. Rostler.The Darwin Model as a Tool for. . . Plasma Simulation.Phys. Fluids, 14:446–448, 1971.
T. B. Krause, A. Apte, and P. J. Morrison.A unified approach to the Darwin approximation.Phys. Plasmas, 14:102112, 2007.
C. W. Nielson and H. R. Lewis.Particle-Code Models in the Nonradiative Limit.In J. Killeen, editor, Meth. Comp. Phys. 16, ControlledFusion
Todd B. Krause Unifying Darwin