Perturbation Methods in Classical Mechanics

Lecture 2.A - Perturbation Methods in Classical Mechanics

Alain J. Brizard (Saint Michael’s College)

Adventures in the Land of Gyrokinetics Department of Applied Physics/ Fusion and Plasma Physics Aalto University, Espoo (Finland) May 1-12, 2017

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.A - Perturbation Methods in Classical Mechanics OUTLINE

1. Lie-transform Perturbation Methods

2. Geometric Interpretation of Hamiltonian Perturbation Theory

3. Hamiltonian & Lagrangian Perturbation Theories

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.A - Perturbation Methods in Classical Mechanics 1. Lie-transform Perturbation Methods

Dynamical reduction of Lagrangian & Hamiltonian mechanics is carried out by near-identity transformations ( 1):

T = ···T2 T1 : z ≡ Z0 → Z1 → Z2 → · · · → z

a. Near-identity transformation T : z → z(z; ) ≡ Tz

1 ∂G a za(z, ) = za + G a + 2 G a + G b 1 + ··· 1 2 2 1 ∂zb

n a a is generated by vector ﬁelds Gn : Tn ≡ exp( Gn ∂/∂z ).

−1 −1 ◦ Inverse transformation T : z → z(z; ) ≡ T z 1 ∂G a za(z, ) = za − G a − 2 G a − G b 1 + ··· 1 2 2 1 ∂zb

is generated by the same vector ﬁelds (G1, G2, ··· ).

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.A - Perturbation Methods in Classical Mechanics b. Pull-back and push-forward operations

◦ Pull-back operation T : f → f ≡ Tf = (T1 T2 ··· ) f

−1 −1 −1 −1 ◦ Push-forward operation T : f → f ≡ T f = (··· T2 T1 ) f

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.A - Perturbation Methods in Classical Mechanics c. Lie-transform deﬁnitions of pull-back and push-forward

n −1 n Tn ≡ exp ( LGn ) and Tn ≡ exp (− LGn )

◦ Since a Lie derivative LG preserves tensor nature

ωk (k − form) → LGωk ≡ ıG · (dωk ) + d(ıG · ωk )

then so do the pull-back and push-forward operators.

d. Jacobian for transformed phase-space coordinates ∂ J d6z ≡ T−1(J d6z) → J = J − J G a + ··· ∂za 1 Exercise: Derive 2nd -order terms

a a ◦ Canonical transformation: Gn = {Sn, z } and J = J .

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.A - Perturbation Methods in Classical Mechanics e. Lie-transform pull-back and push-forward operators

◦ Pull-back operator T ≡ T1 T2 T3 ··· 1 1 T = 1+ L +2 L + L2 +3 L + L L + L3 +··· 1 2 2 1 3 1 2 6 1

−1 −1 −1 −1 ◦ Push-forward operator T ≡ · · · T3 T2 T1 1 1 T−1 = 1− L −2 L − L2 −3 L − L L + L3 +··· 1 2 2 1 3 2 1 6 1

−1 −1 rd Exercise: Show that T (T f ) = f = T(T f ) up to 3 order

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.A - Perturbation Methods in Classical Mechanics 2. Geometric Interpretation of Hamiltonian Perturbation Theory

a. Multi-Hamiltonian dynamics  H, d/dt = ∂/∂t + { , H} (time evolution) i  (Si , σ ) ≡  S, d/d = ∂/∂ + { , S} (perturbation evolution)

◦ Variational Principle δAC [Z] ≡ 0 (arbitrary path C) Z h a i i δAC [Z] ≡ δ Γa dZ − Si dσ C Z a b ∂Si i = δZ ωab dZ (t, ) − a dσ C ∂Z ◦ Euler-Lagrange Equations ∂S ∂Za ω dZb ≡ i dσi → = {Za, S } ab ∂Za ∂σi i

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.A - Perturbation Methods in Classical Mechanics b. Path Independence (Perturb ↔ Evolve)

◦ For two paths C and C 0 (with same end points) ◦ Stokes’ theorem (∂D ≡ C 0 − C)(Exercise: Show this) I Z i i AC 0 − AC = Γ − Si dσ ≡ dΓ − dSi ∧ dσ ∂D D Z ∂S ∂H ≡ d ∧ dt − + {S, H} D ∂t ∂

i ◦ Path independence AC ≡ AC 0 ⇒ dΓ = dSi ∧ dσ : ∂S ∂H + {S, H} ≡ ∂t ∂ ◦ Commuting Hamiltonian ﬂows for all F (Exercise: Show this)

d d ∂S ∂H , F ≡ F , − + {S, H} = 0 dt d ∂t ∂

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.A - Perturbation Methods in Classical Mechanics 3. Hamiltonian & Lagrangian Perturbation Theories

a. Canonical Hamiltonian Perturbation Theory

◦ Perturbed Hamiltonian

2 H = H − w ≡ H0 + H1 + H2 + ··· − w

◦ Perturbed nth-order Hamiltonian

Hn ≡ hHni (slow) + Hen (fast) • Canonical near-identity transformation ∂S 1 ∂ G a ≡ {S , za} = n Jba ≡ JS Jba n n ∂zb J ∂zb n −1 6 6 ◦ Transformed Jacobian: T (J d z) ≡ J d z ∂2 J = J − JS Jba + · · · ≡ J ∂za∂zb 1

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.A - Perturbation Methods in Classical Mechanics • Reduced Hamiltonian (hSni ≡ 0, n ≥ 1)

1 n o H ≡ T−1H = H−{S , H}−2 {S , H} − S , {S , H} +··· 1 2 2 1 1

◦ Zeroth-order analysis

H0 ≡ H0 = H0 − w

◦ First-order analysis  H1 ≡ hH1i d0S1  H1 = H1 − ⇒ dt  −1 S1 ≡ (d0/dt) He1

◦ Up to ﬁrst order, with H ≡ (H0 + H1) − w: dza n o za = za + {S , za} + · · · → = za, H + hH i 1 dτ 0 1

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.A - Perturbation Methods in Classical Mechanics • Second-order analysis (d0S1/dt = H1 − H1)

d S 1 d S H = H − 0 2 − {S , H } + S , 0 1 2 2 dt 1 1 2 1 dt d S 1 n o = H − 0 2 − {S , H } + S , (H − H ) 2 dt 1 1 2 1 1 1 d S 1 n o = H − 0 2 − S , (H + H ) 2 dt 2 1 1 1

d0S2 1 n o = hH2i + He2 − − S1, 2 hH1i + He1 dt 2

 1 Dn oE  H2 = hH2i − S1, He1 ≡ hH2i + hK2i  2 ⇒  −1  S2 = (d0/dt) He2 − {S1, hH1i} + Ke2

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.A - Perturbation Methods in Classical Mechanics ◦ Physical meaning of ponderomotive Hamiltonian hK2i 1 Dn oE 1 ∂ ∂S1 hK2i = − S1, H1 = − · H1 2 Θ 2 ∂J ∂Θ Θ

2 ! ∂ X m |hm| = · ∂J ω − m · Ω m where

X h i i hm H = h ei(m ·Θ−ω t) + c.c. → S = 1 m 1m ω − m · Ω m ◦ Wave-particle-resonance Denominator     ω − m · Ω ≡ ω −  mg ωg + mb ωb + md ωd  | {z } | {z } | {z } Gyro Bounce Drift

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.A - Perturbation Methods in Classical Mechanics b. Phase-space Lagrangian Perturbation Theory

◦ Perturbed Symplectic Structure & Hamiltonian

     −1  Γ Γ0 + Γ1 + ··· Γ ≡ T Γ + dσ   =   →   −1 H H0 + H1 + ··· H ≡ T H

  Hamiltonian Formulation : Γ ≡ Γ0 (Γn ≡ 0, n ≥ 1) ⇒  Symplectic Formulation : Γ ≡ Γ0 + Γ1 + ···

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.A - Perturbation Methods in Classical Mechanics • Hamiltonian Formulation – First Order

◦ First-order symplectic structure

0 ≡ Γ1 = Γ1 − L1 Γ0 + dσ1

= Γ1 − ı1 · ω0 + d σ1 − ı1 · Γ0

≡ Γ1 − ı1 · ω0 + dS1

◦ Solution for G1:

b 0 = Γ1a − G1 ω0 ba + ∂aS1

c b ac ac ac G1 ≡ G1 (ω0 ba J0 ) = ∂aS1 J0 + Γ1a J0

c c = {S1, z }0 + g1 (noncanonical part)

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.A - Perturbation Methods in Classical Mechanics ◦ First-order Hamiltonian

H1 ≡ H1 − L1H0 d S = H − g c ∂ H − 0 1 1 1 c 0 dt d za d S = H − Γ 0 − 0 1 1 1a dt dt | {z } ≡ K1  H1 ≡ hK1i d0S1  H1 ≡ K1 − ⇒ dt  −1 S1 ≡ (d0/dt) Ke1

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.A - Perturbation Methods in Classical Mechanics • Hamiltonian Formulation – Second Order

◦ Second-order symplectic structure 1 Γ = Γ − L Γ − L Γ + L2 Γ + dσ 2 2 2 0 1 1 2 1 0 2 1 = Γ − L Γ − L Γ + L (Γ − Γ ) + dσ 2 2 0 1 1 2 1 1 1 2 1 ≡ Γ − ı · ω − ı · ω + dS 2 2 0 2 1 1 2

1 Γ ≡ 0 ⇒ G a = {S , za} + Γ − G c ω Jba 2 2 2 0 2b 2 1 1 cb 0

a a ≡ {S2, z }0 + g2 (non-canonical part)

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.A - Perturbation Methods in Classical Mechanics b ◦ Second-order Hamiltonian (G1 ∂bH0 = H1 − H1) 1 H = H − G a∂ H + G a∂ G b ∂ H − G a ∂ H 2 2 2 a 0 2 1 a 1 b 0 1 a 1 d S 1 = H − 0 2 − g a∂ H + G a∂ H − H − G a ∂ H 2 dt 2 a 0 2 1 a 1 1 1 a 1

d0S2 a ∂H0 1 n o = H2 − − g2 − S1, (H1 + hK1i) dt ∂za 2 0 1 n a o − Γ1a z , (H1 + hK1i) 2 0 Exercise 2.2: Show that

b d0z 1 D n o E H2 = H2 − Γ2b − S1, He1 dt 2 0 b 1 n a o a d0z − Γ1a z , H1 + hK1i − G1 ω1 ab 2 0 dt

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.A - Perturbation Methods in Classical Mechanics Lecture 2.B - Perturbation Methods in Classical Mechanics

Alain J. Brizard (Saint Michael’s College)

Adventures in the Land of Gyrokinetics Department of Applied Physics/ Fusion and Plasma Physics Aalto University, Espoo (Finland) May 1-12, 2017

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.B - Examples OUTLINE

1. Example 1: Perturbed pendulum

2. Example 2: Oscillation-center Theory

3. Reduced Vlasov Equation

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.B - Examples 1. Example 1: Perturbed Pendulum

1 2 1 4 a. Pendulum potential V (q) = 1 − cos q = 2! q − 4! q + ···

• Unperturbed Hamiltonian (simple pendulum)

2 2 H0(q, p) = p /2 + q /2

◦ Action-angle coordinates z = (J, θ): √ q = 2 J sin θ √ → K (z) ≡ H (q(z), p(z)) = J p = 2 J cos θ 0 0

◦ Unperturbed Hamilton’s equations √  q (t) = 2 J sin t ∂K ∂K  0 0 J˙ = − 0 ≡ 0 and θ˙ = 0 ≡ 1 → 0 ∂θ 0 ∂J √  p0(t) = 2 J0 cos t

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.B - Examples • Perturbation Hamiltonian H (q, p) = − q4 → K (J, θ) = − J2 sin4 θ 1 24 1 6

◦ Perturbation is small (i.e., |K1| < K0), if < 6/Jmax.

◦ New Hamiltonian K ≡ K0 + K1 now depends on the angle variable θ, so that the action variable J is no longer invariant: ∂K J˙ = − 1 6= 0 ∂θ

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.B - Examples b. Hamiltonian (canonical) perturbation theory

• Construct new action-angle coordinates z = (J, θ) such that

K(J(z), θ(z)) = K(z) ≡ K 0(J) + K 1(J) + ···

◦ Canonical transformation (J, θ) → (J, θ):

1 n o za = za + {S , za} + 2 {S , za} + S , {S , za} + ··· 1 2 2 1 1

where (S1, S2, ··· ) are generating functions, and the action-angle canonical Poisson bracket is {F , G} = ∂θF ∂J G − ∂J F ∂θG.

• New Hamiltonian 1 n o K = K − {S , K} − 2 {S , K} + S , {S , K} + ··· 1 2 2 1 1

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.B - Examples • When the original Hamiltonian is

2 K = K0(J) + K1(J, θ) + K2(J, θ) + ···

the transformed Hamiltonian is

2 K = K 0(J) + K 1(J) + K 2(J) + ···

◦ Lie-transform relations

K 0 = K0 ∂S K = K − {S , K } = K − 1 1 1 1 0 1 ∂θ 1 n o K = K − {S , K } − {S , K } + S , {S , K } 2 2 2 0 1 1 2 1 1 0 ∂S 0 ∂S 1 ∂S = K − 2 − K (J) 1 − S , 1 2 ∂θ 1 ∂θ 2 1 ∂θ

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.B - Examples c. Perturbed pendulum h i K(J, θ) = J − J2 3 − 4 cos 2θ − cos 4θ 48

◦ At zeroth order, we easily ﬁnd K 0 = J.

◦ At ﬁrst order, we impose

2 K 1(J) ≡ hK1(J, θ)i = − J /16

2 ∂S1/∂θ ≡ Ke1 ≡ J 4 cos 2θ − cos 4θ /48

◦ First-order generating function

J2 S (J, θ) = 8 sin 2θ − sin 4θ 1 192

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.B - Examples • Up to ﬁrst order, new action-angle coordinates (J, θ)

∂S J = J + 1 = J + J2 4 cos 2θ − cos 4θ ∂θ 48 ∂S θ = θ − 1 = θ − J 8 sin 2θ − sin 4θ ∂J 96 Exercise: Show that 1 J = p2 + q2 − 5 q4 − 6 p2q2 − 3 p4 2 192 and verify that, using the perturbed-pendulum equations  q˙ = p  → J˙ = O(2) p˙ = − q + q3/6 

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.B - Examples Plots of the old action J(t) (black) and the new actions J(t) (with ﬁrst-order corrections = blue and second-order corrections = red) corresponding to the initial conditions q(0) = 2 and p(0) = 0 with perturbation parameter = 0.25.

2.00

1.95

1.90

1.85

0 2 4 6 8 10

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.B - Examples d. New Hamilton’s equations of motion

∂K ∂K J˙ = − = 0 and θ˙ = = 1 − J ≡ Ω ∂θ ∂J 8 ◦ Inverse action-angle transformation

2 J(t) = J + J cos 4(θ + Ωt) − 4 cos 2(θ + Ωt) 48 0 0 θ(t) = θ + Ωt − J sin 4(θ + Ωt) − 8 sin 2(θ + Ωt) 0 96 0 0

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.B - Examples ◦ Original pendulum coordinates

q(t) = p2J(t) sin θ(t) and p(t) = p2J(t) cos θ(t)

q(t) 2

q(approximate) 1

q(exact) Time t 0 2 4 6 8 10

−1

−2

◦ Note that approximate solution is excellent for t < −1

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.B - Examples e. Second-order analysis J ∂S1 1 ∂S1 ∂S2 K 2 = − S1, − 8 ∂θ 2 ∂θ ∂θ " 2# 1 ∂ ∂S1 ∂ J 1 ∂S1 ∂S1 = − − S2 − S1 − 2 ∂J ∂θ ∂θ 8 2 ∂J ∂θ

◦ At second order, we impose

2 1 ∂h(Ke1) i K 2 ≡ − 2 ∂J J 1 ∂S1 ∂S1 ∂S1 ∂S1 S2 ≡ S1 + − 8 2 ∂J ∂θ ∂J ∂θ ◦ New action variable ∂S ∂S 1 ∂S J = J + 1 + 2 2 + S , 1 + · · · → J˙ = O(3) ∂θ ∂θ 2 1 ∂θ

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.B - Examples • Pendulum action up to second order in (Exercise)

∂S ∂S 1 ∂S J = J + 1 + 2 2 + S , 1 ∂θ ∂θ 2 1 ∂θ ∂S J ∂S = J + 1 + 2 1 ∂θ 8 ∂θ " # 1 16 + 2 p2 + q2 = p2 + q2 + 3p4 + 6 p2q2 − 5q4 2 16 · 192

◦ Using the perturbed-pendulum equations

p q3 J˙ = 3 9p4 + 18 p2q2 + q4 9216

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.B - Examples 2. Oscillation-center Theory

a. Paradigm: Charged particles interacting with a high-frequency, short-wavelength electromagnetic wave

• Expansion in powers of wave amplitude

∞ Φ X Φ E = n n → A An B n=0 where

(Φ0, A0) ≡ lowest-order background plasma

(Φ1, A1) ≡ primary (external) wave ﬁelds

(Φ2, A2) ≡ second-order (back-reaction) ﬁelds

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.B - Examples −1 −1 • Eikonal Representation (0 ∇Θ = k, 0 ∂t Θ = − ω)

◦ Weak-background-ﬁeld (WBF) ordering (0 1)

Φ0 ≡ Φ0(0r, 0t) and A0 ≡ A0(0r, 0t)

E0 ≡ 0 E0(0r, 0t) and B0 ≡ 0 B0(0r, 0t)

◦ Eikonal representation for ﬁrst-order wave ﬁelds

! Φ1 Φe1 −1 ≡ exp i0 Θ(0r, 0t) + c.c. A1 Ae1

◦ Eikonal representation for second-order wave ﬁelds

! Φ2 Φ2 Φe2 ≡ + e2i Θ/0 + c.c. A2 A2 Ae2

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.B - Examples • Perturbed Symplectic Structure & Hamiltonian

2 ◦ Unperturbed Hamiltonian H ≡ |p| /2m − w ≡ H0

◦ Perturbed symplectic structure h e i Γ = p + A + A + 2 A + ··· · dx c 0 1 2 2 − w + e Φ0 + Φ1 + Φ2 + ··· dt

−1 ◦ Unperturbed Poisson bracket (Γ ≡ T Γ + dσ ≡ Γ0) ∂F ∂G ∂F ∂G e ∂F ∂G {F , G}0 = µ − µ + F(0)µν ∂x ∂pµ ∂pµ ∂x c ∂pµ ∂pν

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.B - Examples b. First-order Lie-transform Perturbation Analysis

v d S d S 0 = H = e Φ − · A − 0 1 ≡ K − 0 1 1 1 c 1 dt 1 dt 0 ◦ Eikonal equation for Se1 (ω = ω − k · v):

0 v ∂Se1 −i ω Se1 + 0 e E0 + × B0 · = Ke1 c ∂p

◦ WBF-eikonal expansion → Se1 = Se10 + 0 Se11 + ···

Ke1 ie v Se10 = i and Se11 = − E0 + × B0 · eξ ω0 ω0 c ◦ First-order displacement ξ (Exercise):

∂Se10 e v eξ ≡ = − Ee1 + × Be1 ∂p mω02 c

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.B - Examples c. Second-order Lie-transform Perturbation Analysis

◦ Second-order Hamiltonian v d S e ∂S v H ≡ e Φ − · A − 0 2 + 1 · E + × B 2 2 c 2 dt 2 ∂p 1 c 1

◦ Eikonal-averaged second-order Hamiltonian (Exercise)

v 02 2 H2 = e Φ2 − · A2 + mω |eξ| c h v i − E · π + B · µ + π × 0 0 2 0 2 2 c ◦ Ponderomotive Hamiltonian e ∂S10 v 02 2 · E1 + × B1 = m ω |eξ| 2 ∂p c

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.B - Examples ◦ WBF Ponderomotive Hamiltonian correction (Exercise)

e ∂S v v 11 · E + × B = − π · E + × B 2 ∂p 1 c 1 2 0 c 0

− µ2 · B0

◦ Ponderomotive electric dipole moment

∗ −1 ∂H2 π2 ≡ e k × i eξ × eξ = − 0 ∂E0 ◦ Ponderomotive magnetic dipole moment e 0 ∗ −1 ∂H2 v ∂H2 µ2 ≡ ω i eξ × eξ = − 0 + × c ∂B0 c ∂E0

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.B - Examples d. Push-forward Derivation

−1 ◦ Reduced displacement ρ ≡ T x − x: 2 e ∂ξ ρ ≡ ξ + ξ · ∇ξ − ∇S + A · − 2 G x + ··· 2 1 c 1 ∂p 2

◦ Eikonal-averaged reduced displacement

2 ∗ 2 −1 ρ = k × −i eξ × eξ ≡ e π2

◦ Ponderomotive magnetization vector

d x e d ρ e ρ × = 2 π × v and ρ × = 2 c µ dt 2 2 dt 2

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.B - Examples e. Oscillation-center Transformation

∂S x = x − 10 + ··· = x − ξ + ··· ∂p e d ξ p = p + ∇S + A + ··· = p − m 0 + ··· 10 c 1 dt ∂S d ξ w = w − 10 − e Φ + ··· = w − mv · 0 + ··· ∂t 1 dt

d x |p|2 p = 0 and w = dt 2m

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.B - Examples 3. Reduced Vlasov Equation

a. Reduced Vlasov Equation in 6D-space

d F ∂F 0 = ≡ + F , H dt ∂t • Near-identity Canonical Transformation

1 n o zα ≡ zα + {S , zα} + 2 {S , zα} + S , {S , zα} + ··· 1 2 2 1 1

• Reduced Vlasov Operator

d F d ∂ F n o ≡ T−1 T F = + F , T−1H dt dt ∂t

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.B - Examples ◦ Reduced time derivative ∂ F ∂ ≡ T−1 T F ∂t ∂t ∂F ∂S ∂S 2 ∂S = + 1 + 2 2 + 1 , S + ··· , F ∂t ∂t ∂t 2 ∂t 1

◦ Push-forward of Hamiltonian in 6D-space

1 n o T−1H = H − {S , H}−2 {S , H} − S , {S , H} +··· 1 2 2 1 1

◦ Reduced Hamiltonian dS dS 1 dS H ≡ H − 1 − 2 2 − S , 1 + ··· dt dt 2 1 dt

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.B - Examples b. Iterative Solution of the Vlasov Equation

◦ Unperturbed Vlasov Equation (known solution) ∂F 0 + {F , H } = 0 ∂t 0 0 ◦ Perturbative Pull-back F0 → F = F0 + F1 + ···

F ≡ TF0 = F0 + {S1, F0} 1 n o + 2 {S , F } + S , {S , F } + ··· 2 0 2 1 1 0

◦ Perturbative Hamiltonian H0 → H = H0 + H1 + ··· d S d S 1 d S H ≡ H + 0 1 + 2 0 2 + S , 0 1 + ··· 0 dt dt 2 1 dt where d S 0 1 = H dt 1 d S 1 d S 1 n o 0 2 = H − S , 0 1 = H − S , H dt 2 2 1 dt 2 2 1 1

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.B - Examples ◦ Lie-transform equation (S ≡ S1 + 2 S2 + ··· ) ∂S n o ∂H + S, H = = H + 2 H + ··· ∂t ∂ 1 2 ◦ Iterative Solution ∂F 0 = + {F , H} ∂t ∂F d S = T 0 + {F , H } + 0 1 − H , F + ··· ∂t 0 0 dt 1 0 ∂F = T 0 + {F , H } ∂t 0 0

∂F ∂F 0 = + {F , H} ≡ T 0 + {F , H } ∂t ∂t 0 0

◦ Pull-back generates solution F = TF0.

Alain Brizard (SMC) - Adventures in the Land of Gyrokinetics Lecture 2.B - Examples