Shape Dynamics of Point Vortices Tomoki Ohsawa April Fools Day, - - PowerPoint PPT Presentation

Shape Dynamics of Point Vortices

Tomoki Ohsawa April Fools’ Day, 2019

Tomoki Ohsawa (UT–Dallas) Georgia Tech 1 / 46

Dynamics of Hurricanes?

Source: U. Washington News & NOAA

Tomoki Ohsawa (UT–Dallas) Georgia Tech 2 / 46

Dynamics of Hurricanes?

Source: U. Washington News & NOAA

Tomoki Ohsawa (UT–Dallas) Georgia Tech 2 / 46

Point Vortex on R2

Point vortex with circulation Γ at x0 = (x0, y0)

• Vorticity ξ(x) = curl u(x) = Γ δ(x − x0)

Γ > 0 x0

• Tomoki Ohsawa (UT–Dallas)

Georgia Tech 3 / 46

Point Vortex on R2

Point vortex with circulation Γ at x0 = (x0, y0)

• Vorticity ξ(x) = curl u(x) = Γ δ(x − x0)

⇓ Velocity field u(x) = Γ 2πr2 (−(y − y0), x − x0)

Γ > 0 x0

• Tomoki Ohsawa (UT–Dallas)

Georgia Tech 3 / 46

Dynamics of N Point Vortices on R2

x1 x2 x3 Γ1 > 0 Γ3 > 0 Γ2 < 0

Each point vortex j located at xj ∈ R2 is convected by the net velocity of the other vortices: ˙ xj(t) =

• 1≤k≤N

k=j

uk(xj(t)),

Tomoki Ohsawa (UT–Dallas) Georgia Tech 4 / 46

Dynamics of N Point Vortices on R2

which gives: ˙ xj = − 1 2π

• 1≤k≤N

k=j

Γk yj − yk xj − xk2 , ˙ yj = 1 2π

• 1≤k≤N

k=j

Γk xj − xk xj − xk2 ,

• r, by setting qj := xj + iyj ∈ C,

˙ qj = i 2π

• 1≤k≤N

k=j

Γk qj − qk |qj − qk|2 for j ∈ {1, . . . , N}.

Tomoki Ohsawa (UT–Dallas) Georgia Tech 5 / 46

What is Hamiltonian System?

Classical Hamiltonian system

q = position, p = m ˙ q = momentum Hamiltonian H(q, p) = p2 2m

• Kinetic Energy

+ V (q)

Potential Energy

= Total Energy The Hamiltonian system [ ˙ q ˙ p]

• I

−I

• =

∂H ∂q ∂H ∂p

• ⇐

⇒ ˙ q = ∂H ∂p , ˙ p = −∂H ∂q gives the equations of motion (Newton’s Second Law): md2q dt2 = −∇V (q).

Tomoki Ohsawa (UT–Dallas) Georgia Tech 6 / 46

Symplectic Geometry and Hamiltonian Systems

Symplectic manifold (“Phase space”) P = {(q, p)} equipped with symplectic form Ω = dq ∧ dp Vector field XH = ˙ q ∂ ∂q + ˙ p ∂ ∂p Contraction iXHΩ = Ω(XH, ·) = − ˙ p dq + ˙ q dp Exterior differential dH = ∂H ∂q dq + ∂H ∂p dp Hamilton’s Eq. iXHΩ = dH = ⇒ ˙ q = ∂H ∂p , ˙ p = −∂H ∂q

Tomoki Ohsawa (UT–Dallas) Georgia Tech 7 / 46

Symplectic Geometry and Hamiltonian Systems

More generally...

Definition (Hamiltonian System on Symplectic Manifold)

Manifold P Symplectic form Ω (closed non-degenerate two-form) Hamiltonian H : P → R

P Tz3P T

z1

P Tz2P z3 z2 z1

Tomoki Ohsawa (UT–Dallas) Georgia Tech 8 / 46

Symplectic Geometry and Hamiltonian Systems

More generally...

Definition (Hamiltonian System on Symplectic Manifold)

Manifold P Symplectic form Ω (closed non-degenerate two-form) Hamiltonian H : P → R

P T

z3

P T

z1

P T

z

2P

z3 z2 z1 XH(z1) XH(z2) XH(z3)

Define a Hamiltonian system by iXHΩ = dH = ⇒ Determines the vector field (dynamics) XH on P.

Tomoki Ohsawa (UT–Dallas) Georgia Tech 8 / 46

Symplectic Geometry and Hamiltonian Systems

More generally...

Definition (Hamiltonian System on Symplectic Manifold)

Manifold P Symplectic form Ω (closed non-degenerate two-form) Hamiltonian H : P → R

P T

z3

P T

z1

P T

z

2P

z3 z2 z1 XH(z1) XH(z2) XH(z3)

Define a Hamiltonian system by iXHΩ = dH = ⇒ Determines the vector field (dynamics) XH on P.

Tomoki Ohsawa (UT–Dallas) Georgia Tech 8 / 46

Hamiltonian Dynamics of N Point Vortices on R2

Symplectic form on CN = {(q1, . . . , qN)}: Ω =

N

• j=1

ΓjΩj with Ωj := −1 2 Im(dqj ∧ dq∗

j ) = dxj ∧ dyj

Hamiltonian H : CN → R H(q1, . . . , qN) = − 1 4π

• 1≤j<k≤N

ΓjΓk ln |qj − qk|2. Then the Hamiltonian system iXHΩ = dH gives the equations for point vortices: For j ∈ {1, . . . , N}. ˙ qj = i 2π

• 1≤k≤N

k=j

Γk qj − qk |qj − qk|2

Tomoki Ohsawa (UT–Dallas) Georgia Tech 9 / 46

Dynamics of 3 Point Vortices on R2

Tomoki Ohsawa (UT–Dallas) Georgia Tech 10 / 46

Dynamics of 3 Point Vortices on R2

Tomoki Ohsawa (UT–Dallas) Georgia Tech 11 / 46

Dynamics of 4 Point Vortices on R2

Tomoki Ohsawa (UT–Dallas) Georgia Tech 12 / 46

Shape Dynamics of Vortices on R2

Distance between vortices: ℓjk := |qj − qk| Shape dynamics or Equations of relative motion: d dt ℓ2

ij = 2

π

• 1≤l≤N

i=j=k

ΓkAijk

• 1

ℓ2

jk

− 1 ℓ2

ki

• ,

A123 Γ1 Γ2 Γ3

12 23 31

where Aijk = signed area of triangle defined by vortices (i, j, k)

Tomoki Ohsawa (UT–Dallas) Georgia Tech 13 / 46

Shape Dynamics of Vortices on R2

Distance between vortices: ℓjk := |qj − qk| Shape dynamics or Equations of relative motion: d dt ℓ2

ij = 2

π

• 1≤l≤N

i=j=k

ΓkAijk

• 1

ℓ2

jk

− 1 ℓ2

ki

• ,

A123 Γ1 Γ2 Γ3

12 23 31

where Aijk = signed area of triangle defined by vortices (i, j, k)

Goal

Is shape dynamics also Hamiltonian? Geometric structure behind it?

Tomoki Ohsawa (UT–Dallas) Georgia Tech 13 / 46

Background: Lie Groups and Lie Algebras

G = Lie group “continuous transformations” g := TeG = Lie algebra of G “infinitesimal transformations” g∗ = dual of Lie algebra g

e G g : = T

e

G

Tomoki Ohsawa (UT–Dallas) Georgia Tech 14 / 46

Background: Lie Groups and Lie Algebras

G = Lie group “continuous transformations” g := TeG = Lie algebra of G “infinitesimal transformations” g∗ = dual of Lie algebra g

e G g : = T

e

G

Example: 3D rotation group SO(3)

Rotations: G = SO(3) =

• R ∈ R3×3 | RTR = I, det R = 1
• Angular velocities:

g = so(3) = TISO(3) =

• ξ ∈ R3×3 | ξT = −ξ
• ∼

= R3

Tomoki Ohsawa (UT–Dallas) Georgia Tech 14 / 46

SE(2)-Action on the Plane R2

Lie group SE(2) := SO(2) ⋉ R2 = Rθ a 1

• | Rθ =

cos θ − sin θ sin θ cos θ

• , θ ∈ [0, 2π), a ∈ R2
• = All rotations of R2 and translations of R2 combined

SE(2)-action on R2:

x

Tomoki Ohsawa (UT–Dallas) Georgia Tech 15 / 46

SE(2)-Action on the Plane R2

Lie group SE(2) := SO(2) ⋉ R2 = Rθ a 1

• | Rθ =

cos θ − sin θ sin θ cos θ

• , θ ∈ [0, 2π), a ∈ R2
• = All rotations of R2 and translations of R2 combined

SE(2)-action on R2:

x θ Rθx

Tomoki Ohsawa (UT–Dallas) Georgia Tech 15 / 46

SE(2)-Action on the Plane R2

Lie group SE(2) := SO(2) ⋉ R2 = Rθ a 1

• | Rθ =

cos θ − sin θ sin θ cos θ

• , θ ∈ [0, 2π), a ∈ R2
• = All rotations of R2 and translations of R2 combined

SE(2)-action on R2:

x θ Rθx a Rθx + a

Tomoki Ohsawa (UT–Dallas) Georgia Tech 15 / 46

SE(2)-Symmetry of N Point Vortices

The Hamiltonian H is invariant under the SE(2)-action. = ⇒ The system of N point vortices is invariant under the SE(2)-action. θ a

1 1 2 2 3 3 1 2 3

These configurations are essentially the same because the shape of the vortices is the same.

Tomoki Ohsawa (UT–Dallas) Georgia Tech 16 / 46

SE(2)-Symmetry of N Point Vortices

The Hamiltonian H is invariant under the SE(2)-action. = ⇒ The system of N point vortices is invariant under the SE(2)-action. θ a

1 1 2 2 3 3 1 2 3

These configurations are essentially the same because the shape of the vortices is the same. = ⇒ After “dividing” it by SE(2), all that matters is the shape of point vortices.

Tomoki Ohsawa (UT–Dallas) Georgia Tech 16 / 46

Background: Noether’s Theorem

Suppose G Lie group and g its Lie algebra G acts on a symplectic manifold P in a canonical manner Then one may define a corresponding map J: P → g∗ called a momentum map.

Theorem (Noether)

If a Hamiltonian system on P has a G-symmetry, then J is a conserved quantity of the Hamiltonian system.

Tomoki Ohsawa (UT–Dallas) Georgia Tech 17 / 46

Simple Example: A Free Particle on the Plane

Configuration space R2 = {x = (x1, x2)} Symplectic manifold T ∗R2 = {(x, p) = (x1, x2, p1, p2)} Hamiltonian H : T ∗R2 → R; H = 1 2m(p2

1 + p2 2)

Tomoki Ohsawa (UT–Dallas) Georgia Tech 18 / 46

Simple Example: A Free Particle on the Plane

Configuration space R2 = {x = (x1, x2)} Symplectic manifold T ∗R2 = {(x, p) = (x1, x2, p1, p2)} Hamiltonian H : T ∗R2 → R; H = 1 2m(p2

1 + p2 2)

Example

Symmetry under translations by G = R2: (x1, x2) → (x1 + a, x2 + b). Momentum map J: T ∗R2 → R2; J(x, p) = (p1, p2) = linear momentum

Tomoki Ohsawa (UT–Dallas) Georgia Tech 18 / 46

Simple Example: A Free Particle on the Plane

Configuration space R2 = {x = (x1, x2)} Symplectic manifold T ∗R2 = {(x, p) = (x1, x2, p1, p2)} Hamiltonian H : T ∗R2 → R; H = 1 2m(p2

1 + p2 2)

Example

Symmetry under translations by G = R2: (x1, x2) → (x1 + a, x2 + b). Momentum map J: T ∗R2 → R2; J(x, p) = (p1, p2) = linear momentum

Example

Symmetry under rotations G = SO(2): x → Rx where R ∈ SO(2) Momentum map J: T ∗R2 → so(2)∗ ∼ = R; J(x, p) = xpy − ypx = angular momentum

Tomoki Ohsawa (UT–Dallas) Georgia Tech 18 / 46

Background: Hamiltonian Reduction

Theorem (Marsden & Weinstein)

Hamiltonian dynamics on P with (Lie group) G-symmetry

P

Tomoki Ohsawa (UT–Dallas) Georgia Tech 19 / 46

Background: Hamiltonian Reduction

Theorem (Marsden & Weinstein)

Hamiltonian dynamics on P with (Lie group) G-symmetry and momentum map J

J−1(µ) P

Tomoki Ohsawa (UT–Dallas) Georgia Tech 19 / 46

Background: Hamiltonian Reduction

Theorem (Marsden & Weinstein)

Hamiltonian dynamics on P with (Lie group) G-symmetry and momentum map J

J−1(µ) P Gµ-orbits

Tomoki Ohsawa (UT–Dallas) Georgia Tech 19 / 46

Background: Hamiltonian Reduction

Theorem (Marsden & Weinstein)

Hamiltonian dynamics on P with (Lie group) G-symmetry and momentum map J ⇓ Hamiltonian dynamics on Pµ := J−1(µ)/Gµ

J−1(µ) P Gµ-orbits Pµ

Tomoki Ohsawa (UT–Dallas) Georgia Tech 19 / 46

Background: Hamiltonian Reduction

Theorem (Marsden & Weinstein)

Hamiltonian dynamics on P with (Lie group) G-symmetry and momentum map J ⇓ Hamiltonian dynamics on Pµ := J−1(µ)/Gµ

J−1(µ) P Gµ-orbits Pµ

Main Idea

If we apply this idea to point vortices on P = R2 with G = SE(2), then the reduced dynamics is the shape dynamics.

Tomoki Ohsawa (UT–Dallas) Georgia Tech 19 / 46

Reduction by R2: 1st Stage of SE(2)-Reduction

Translational action by R2 ∼ = C: C × CN → CN; (a, q := (q1, . . . , qN)) → (q1 + a, . . . , qN + a) Momentum map (conserved quantity): I(q) := −i

N

• j=1

Γjqj (“linear impulse”) I is equivariant ⇐ ⇒ total circulation Γ :=

N

• j=1

Γj = 0

Tomoki Ohsawa (UT–Dallas) Georgia Tech 20 / 46

Reduction by R2: 1st Stage of SE(2)-Reduction

Proposition (Reduction by Translational Symmetry)

For any c ∈ C:

1

If Γ = 0, the R2-reduced space is I−1(−ic) ∼ = I−1(0) ∼ = CN−1.

2

If Γ = 0, the R2-reduced space is I−1(−ic)/C ∼ = I−1(0)/C ∼ = CN−2.

Tomoki Ohsawa (UT–Dallas) Georgia Tech 21 / 46

R2-Reduced Space

For Γ = 0, R2-Reduced Space: Z := I−1(0) ∼ = CN−1 = {(z1, . . . , zN−1)}, where zj’s are relative coordinates w.r.t. last vortex: (z1, . . . , zN−1) := (q1 − qN, . . . , qN−1 − qN). Γ1 Γ2

q1 q2 q3 z1 z2 qN ΓN ... ΓN−1 zN−1

Tomoki Ohsawa (UT–Dallas) Georgia Tech 22 / 46

Symplectic Structure on R2-Reduced Space

Proposition

1

If Γ = 0, then the symplectic form on the R2-reduced space Z ∼ = CN−1 is ΩZ = −dΘZ, where ΘZ is the one-form defined as ΘZ := 1

2 Im(z∗Kdz) with

K := 1 Γ      −Γ1(Γ − Γ1) Γ1Γ2 . . . Γ1ΓN−1 Γ2Γ1 −Γ2(Γ − Γ2) . . . Γ2ΓN−1 . . . . . . ... . . . ΓN−1Γ1 ΓN−1Γ2 . . . −ΓN−1(Γ − ΓN−1)      .

2

If Γ = 0 then a similar result on Z ∼ = CN−2 with a similar matrix K0.

Tomoki Ohsawa (UT–Dallas) Georgia Tech 23 / 46

Reduction by SO(2): 2nd Stage of SE(2)-Reduction

SO(2) ∼ = S1-action on CN−1: S1 × CN−1 → CN−1;

• eiθ, z = (z1, . . . , zN−1)
• →
• eiθz1, . . . , eiθzN−1
• .

Momentum map (conserved quantity): R(z) = −1 2z∗Kz (“angular impulse”), where K is a non-singular matrix depending on {Γj}N

j=1.

Reduced space R−1(c0)/S1; this is where the shape dynamics is.

Tomoki Ohsawa (UT–Dallas) Georgia Tech 24 / 46

Reduction by SO(2): 2nd Stage of SE(2)-Reduction

SO(2) ∼ = S1-action on CN−1: S1 × CN−1 → CN−1;

• eiθ, z = (z1, . . . , zN−1)
• →
• eiθz1, . . . , eiθzN−1
• .

Momentum map (conserved quantity): R(z) = −1 2z∗Kz (“angular impulse”), where K is a non-singular matrix depending on {Γj}N

j=1.

Reduced space R−1(c0)/S1; this is where the shape dynamics is.

Problem

R−1(c0)/S1 is a rather awkward space to work with. For example, if K > 0 then R−1(c0)/S1 ∼ = P(CN−1) etc. Alternative description of R−1(c0)/S1?

Tomoki Ohsawa (UT–Dallas) Georgia Tech 24 / 46

SO(2)-Reduction by Dual Pair

Two Lie groups acting on CN−1 in a canonical manner: U(K)

not symmetry

CN−1 S1

• symmetry

where U(K) :=

• U ∈ C(N−1)×(N−1) | U∗KU = K
• with its Lie algebra

u(K) :=

• ξ ∈ C(N−1)×(N−1) | ξ∗K + Kξ = 0
• Tomoki Ohsawa (UT–Dallas)

Georgia Tech 25 / 46

SO(2)-Reduction by Dual Pair

Two Lie groups acting on CN−1 in a canonical manner: U(K)

not symmetry

CN−1 S1

• symmetry

where U(K) :=

• U ∈ C(N−1)×(N−1) | U∗KU = K
• with its Lie algebra

u(K) :=

• ξ ∈ C(N−1)×(N−1) | ξ∗K + Kξ = 0
• Pair of momentum maps:

u(K)∗ CN−1 R

not conserved J R conserved

Tomoki Ohsawa (UT–Dallas) Georgia Tech 25 / 46

SO(2)-Reduction by Dual Pair

Two Lie groups acting on CN−1 in a canonical manner: U(K)

not symmetry

CN−1 S1

• symmetry

where U(K) :=

• U ∈ C(N−1)×(N−1) | U∗KU = K
• with its Lie algebra

u(K) :=

• ξ ∈ C(N−1)×(N−1) | ξ∗K + Kξ = 0
• Pair of momentum maps:

u(K)∗ CN−1 R

not conserved J R conserved

They form a dual pair, which implies shape dynamics in R−1(c)/S1 is Lie–Poisson dynamics in u(K)∗

Tomoki Ohsawa (UT–Dallas) Georgia Tech 25 / 46

Background: Lie–Poisson Equation

Given Hamiltonian h: g∗ → R on the dual g∗ of a Lie algebra g, one can formulate the Lie–Poisson equation ˙ µ = ± ad∗

δh/δµ µ

using a natural Poisson bracket on g∗.

Tomoki Ohsawa (UT–Dallas) Georgia Tech 26 / 46

Background: Lie–Poisson Equation

Given Hamiltonian h: g∗ → R on the dual g∗ of a Lie algebra g, one can formulate the Lie–Poisson equation ˙ µ = ± ad∗

δh/δµ µ

using a natural Poisson bracket on g∗.

Example: SO(3) and rigid body dynamics

µ = body angular momentum ∈ so(3)∗ h = rotational energy Lie–Poisson equation ˙ µ = µδh δµ − δh δµµ gives Euler’s equation for rigid body.

X R

x

reference configuration current configuration

B

R(B)

Tomoki Ohsawa (UT–Dallas) Georgia Tech 26 / 46

Main Result

Theorem

1

The shape dynamics of N point vortices with non-vanishing angular impulse is governed by a Lie–Poisson equation in u(K)∗: ˙ µ = − ad∗

δh/δµ µ = −µδh

δµK−1 + K−1 δh δµµ, where µ := J(z) and h is collective Hamiltonian, i.e., H = h ◦ J.

2

Cj(µ) := tr((i Kµ)j) is a Casimir (conserved quantity) for any j ∈ N.

Tomoki Ohsawa (UT–Dallas) Georgia Tech 27 / 46

Example: N = 3 and Γ = 0

Three point vortices with (Γ1, Γ2, Γ3) = (5, 10, 15) Γ :=

3

• j=1

Γj = 0 (q1(0), q2(0), q3(0)) = (1 − 2i, 2 + 4i, −5/3 − 2i)

Tomoki Ohsawa (UT–Dallas) Georgia Tech 28 / 46

Shape Dynamics for N = 3

˙ µ = − ad∗

δh/δµ µ

with µ = i

• µ2

µ3 + i µ4 µ3 − i µ4 µ1

• ∈ u(K)∗ ∼

= R4 where µ1 = ℓ2

23

µ2 = ℓ2

31

µ3 + i µ4 = ℓ13ℓ23eiθ3 x1 x2 x3

31 23 12

θ3

are the shape variables.

Tomoki Ohsawa (UT–Dallas) Georgia Tech 29 / 46

Conserved Quantities

N = 3 and Γ = 0

Hamiltonian h Linear Casimir C1 (essentially the angular impulse R) Quadratic Casimir C2 (apparently) new! C2 =

• (i,j,k)∈Z3
• l4

jk

Γ2

i

+ (l2

ij − l2 jk + l2 ki)2

2ΓjΓk

• + 8Γ1 + Γ2 + Γ3

Γ1Γ2Γ3 A2

123.

where Z3 := {(1, 2, 3), (2, 3, 1), (3, 1, 2)}.

Tomoki Ohsawa (UT–Dallas) Georgia Tech 30 / 46

Periodicity of Shape Dynamics: N = 3 and Γ = 0

Casimir C1 is linear = ⇒ Its level set defines a 3D affine subspace in u(K)∗ ∼ = R4. Green = Level set of quadratic Casimir C2 (green)—ellipsoid here Orange = Level set of Hamiltonian h Shape dynamics is periodic!

Tomoki Ohsawa (UT–Dallas) Georgia Tech 31 / 46

Example: N = 4 and Γ = 0

Recall: The R2-reduced space is Z =

• CN−1

Γ = 0, CN−2 Γ = 0. So N = 4 with Γ = 0 is essentially the same as N = 3 with Γ = 0. Four point vortices with (Γ1, Γ2, Γ3, Γ4) = (5, 10, −7, −8) Γ :=

4

• j=1

Γj = 0 (q1(0), q2(0), q3(0), q4(0)) = (1 − 2i, 2 + 4i, 5i, (25 − 5i)/8)

Tomoki Ohsawa (UT–Dallas) Georgia Tech 32 / 46

Periodicity of Shape Dynamics: N = 4 and Γ = 0

Conserved quantities are essentially the same. Green = Level set of quadratic Casimir C2 (green)—ellipsoid here Orange = Level set of Hamiltonian h Shape dynamics is periodic again!

Tomoki Ohsawa (UT–Dallas) Georgia Tech 33 / 46

From R2 to S2

Shape dynamics of N point vortices on S2

Can we do the same thing on S2?

Tomoki Ohsawa (UT–Dallas) Georgia Tech 34 / 46

Hamiltonian Dynamics of N Point Vortices on S2

Let S2 be the two-sphere with radius R centered at the origin. Symplectic form on (S2)N = {(x1, . . . , xN)}: Ω =

N

• j=1

ΓjΩj with Ωj := area form on S2 Hamiltonian H : (S2)N → R H(x1, . . . , xN) = − 1 4πR2

• 1≤j<k≤N

ΓjΓk ln |xj − xk|2. Then the Hamiltonian system iXHΩ = dH gives ˙ xj = 1 4πR

• 1≤k≤N

k=j

Γk xk × xj R2 − xj · xk .

Tomoki Ohsawa (UT–Dallas) Georgia Tech 35 / 46

Shape Dynamics of Vortices on S2

Euclidean distance between vortices: ℓjk := xj − xk Shape dynamics or equations of relative motion: d dt ℓ2

ij =

1 πR

• 1≤l≤N

i=j=k

ΓkVijk

• 1

ℓ2

jk

− 1 ℓ2

ki

• ,

where Vijk := xi · (xj × xk).

x1 x2 x3 Γ1 Γ2 Γ3

12 23 31

Tomoki Ohsawa (UT–Dallas) Georgia Tech 36 / 46

Shape Dynamics and Reduction: Vortices on S2

Symmetry group: SO(3) Momentum map J: (S2)N → so(3)∗ Reduced space J−1(µ)/SO(3)µ...?

x1 x2 x3 Γ1 Γ2 Γ3

12 23 31

Tomoki Ohsawa (UT–Dallas) Georgia Tech 37 / 46

Lifting Dynamics from S2 to C2—Dynamics

Main Idea

Lift the dynamics from S2 to C2 first and then apply reduction by rotation

Tomoki Ohsawa (UT–Dallas) Georgia Tech 38 / 46

Lifting Dynamics from S2 to C2—Dynamics

Main Idea

Lift the dynamics from S2 to C2 first and then apply reduction by rotation Symplectic form on (C2)N ∼ = C2×N ∋ Φ = [ϕ1 . . . ϕN]: ΩC2 = −dΘC2 with ΘC2 := −2 Im (tr (DΓΦ∗dΦ)) where DΓ := diag(Γ1, . . . , ΓN)

Tomoki Ohsawa (UT–Dallas) Georgia Tech 38 / 46

Lifting Dynamics from S2 to C2—Dynamics

Main Idea

Lift the dynamics from S2 to C2 first and then apply reduction by rotation Symplectic form on (C2)N ∼ = C2×N ∋ Φ = [ϕ1 . . . ϕN]: ΩC2 = −dΘC2 with ΘC2 := −2 Im (tr (DΓΦ∗dΦ)) where DΓ := diag(Γ1, . . . , ΓN) Hamiltonian HC2 := − 1 4π

• 1≤i<j≤N

ΓiΓj ln

• ϕi2 + ϕj22

− 4| ϕi, ϕj |2

• Tomoki Ohsawa (UT–Dallas)

Georgia Tech 38 / 46

Lifting Dynamics from S2 to C2—Dynamics

Main Idea

Lift the dynamics from S2 to C2 first and then apply reduction by rotation Symplectic form on (C2)N ∼ = C2×N ∋ Φ = [ϕ1 . . . ϕN]: ΩC2 = −dΘC2 with ΘC2 := −2 Im (tr (DΓΦ∗dΦ)) where DΓ := diag(Γ1, . . . , ΓN) Hamiltonian HC2 := − 1 4π

• 1≤i<j≤N

ΓiΓj ln

• ϕi2 + ϕj22

− 4| ϕi, ϕj |2

• Hamiltonian system iXHC2 ΩC2 = dHC2 in (C2)N yields Schr¨
• dinger-like

equations: Γj ˙ ϕj = − i 2 ∂HC2 ∂ϕ∗

j

for j ∈ {1, . . . , N}.

Tomoki Ohsawa (UT–Dallas) Georgia Tech 38 / 46

Lifting Dynamics from S2 to C2—Geometry

TN := (S1)N phase symmetry Momentum map J = −2

• Γ1 ϕ12 , . . . , ΓN ϕN2

J−1(−2Γ) ∼ = (S3)N is an invariant submanifold Marsden–Weinstein quotient J−1(−2Γ)/TN = (S2)N (C2)N J−1(−2Γ) ∼ = (S3)N (S2)N

i /TN

Tomoki Ohsawa (UT–Dallas) Georgia Tech 39 / 46

Lifting Dynamics from S2 to C2—Geometry

TN := (S1)N phase symmetry Momentum map J = −2

• Γ1 ϕ12 , . . . , ΓN ϕN2

J−1(−2Γ) ∼ = (S3)N is an invariant submanifold Marsden–Weinstein quotient J−1(−2Γ)/TN = (S2)N Recovers the vortex equations on S2 Γj ˙ ϕj = − i 2 ∂HC2 ∂ϕ∗

j

in C2 ⇓ ˙ xj = 1 4πR

• 1≤k≤N

k=j

Γk xk × xj R2 − xj · xk in S2 (C2)N J−1(−2Γ) ∼ = (S3)N (S2)N

i /TN

Tomoki Ohsawa (UT–Dallas) Georgia Tech 39 / 46

Reduction by U(2)

Action U(2) × C2×N → C2×N; (U, Φ) → UΦ = [Uϕ1 . . . UϕN] ΩC2 and HC2 are both U(2)-invariant SU(2) subgroup action on C2 Rotational action of SO(3) on S2 Momentum map (conserved quantity): K: C2×N → u(2)∗; K(Φ) = −i

N

• i=1

Γiϕiϕ∗

i

Marsden–Weinstein quotient K−1(κ)/U(2)κ for some κ ∈ u(2)∗

Tomoki Ohsawa (UT–Dallas) Georgia Tech 40 / 46

Reduction by U(2)

Action U(2) × C2×N → C2×N; (U, Φ) → UΦ = [Uϕ1 . . . UϕN] ΩC2 and HC2 are both U(2)-invariant SU(2) subgroup action on C2 Rotational action of SO(3) on S2 Momentum map (conserved quantity): K: C2×N → u(2)∗; K(Φ) = −i

N

• i=1

Γiϕiϕ∗

i

Marsden–Weinstein quotient K−1(κ)/U(2)κ for some κ ∈ u(2)∗

Same Problem Again!

What kind of space is K−1(κ)/U(2)κ...?

Tomoki Ohsawa (UT–Dallas) Georgia Tech 40 / 46

U(2)-Reduction by Dual Pair

Two Lie groups acting on C2×N: U(2)

• symmetry

C2×N U(DΓ)

not symmetry

where U(DΓ) =

• V ∈ CN×N | V ∗DΓV = DΓ
• Pair of momentum maps:

u(2)∗ C2×N u(DΓ)∗

conserved K L not conserved

Tomoki Ohsawa (UT–Dallas) Georgia Tech 41 / 46

U(2)-Reduction by Dual Pair

Two Lie groups acting on C2×N: U(2)

• symmetry

C2×N U(DΓ)

not symmetry

where U(DΓ) =

• V ∈ CN×N | V ∗DΓV = DΓ
• Pair of momentum maps:

u(2)∗ C2×N u(DΓ)∗

conserved K L not conserved

They form a dual pair (Skerritt & Vizman [2018]), which implies shape dynamics in K−1(κ)/U(2)κ is a Lie–Poisson dynamics in u(DΓ)∗

Tomoki Ohsawa (UT–Dallas) Georgia Tech 41 / 46

Main Result

Theorem

1

The U(2)-reduced (lifted) dynamics of N point vortices on S2 is governed by a Lie–Poisson equation in u(DΓ)∗: ˙ λ = ad∗

δh/δλ λ = λδh

δλD−1

Γ

− D−1

Γ

δh δλλ, where λ := L(z) and h is collective Hamiltonian, i.e., H = h ◦ L.

2

Cj(λ) := tr((i DΓλ)j) is a Casimir (conserved quantity) for any j ∈ N.

Tomoki Ohsawa (UT–Dallas) Georgia Tech 42 / 46

Remark

The above dynamics is not quite the shape dynamics yet: C2×N (S2)N u(DΓ)∗ J−1(µ)/SO(3)µ

Reduction by TN Reduction by U(2) Reduction by SO(3) Reduction by TN−1

Tomoki Ohsawa (UT–Dallas) Georgia Tech 43 / 46

Remark

The above dynamics is not quite the shape dynamics yet: C2×N (S2)N u(DΓ)∗ J−1(µ)/SO(3)µ

Reduction by TN Reduction by U(2) Reduction by SO(3) Reduction by TN−1

Further reduction by TN−1 yields Poisson structure on shape variables ℓij’s etc.

Tomoki Ohsawa (UT–Dallas) Georgia Tech 43 / 46

Example: N = 3

(Γ1, Γ2, Γ3) = (5, 10, 15) (x1(0), x2(0), x3(0)) = (1, 0, 0), (0, 1, 0), (sin π

4 cos π 3 , sin π 4 sin π 3 , cos π 4 )

Tomoki Ohsawa (UT–Dallas) Georgia Tech 44 / 46

Shape Dynamics on S2: N = 3

˙ λ = ad∗

δh/δµ λ

with λ = − i 2   √ 2λ1 λ12 λ13 ¯ λ12 √ 2λ2 λ23 ¯ λ13 ¯ λ23 √ 2λ3   Diagonal entries are constant: λ1 = λ2 = λ3 = √ 2R T2-action: Using µ12 := λ12¯ λ13λ23 instead of λ12,

• (eiθ1, eiθ2), (µ12, λ13, λ23)
• →
• µ12, eiθ1λ13, eiθ2λ23
• Reduced to (µ12, |λ13|, |λ23|) ∈ C× × R2

+

|λij|2 = 4R2 − ℓ2

ij,

1 2 Im µ12 = V = x1 · (x2 × x3)

x1 x2 x3 Γ1 Γ2 Γ3

12 23 31

Tomoki Ohsawa (UT–Dallas) Georgia Tech 45 / 46

Shape Dynamics on S2: N = 3

Three Point Vortices: N = 3

Casimir C1 is trivially constant; C2 well known; C3 apparently new 3 conserved quantities—Hamiltonian h and Casimirs C2, C3—in C× × R2

+

Three point vortices with Orange: Level set of h Green: Level set of C3 Shape dynamics is periodic again!

Tomoki Ohsawa (UT–Dallas) Georgia Tech 46 / 46