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Moving Frames,Variational Problems, &Geometric Curve Flows

Peter J. Olver

University of Minnesota

http://www.math.umn.edu/∼ olver

Case Western, October, 2009

Basic Notation

x = (x1, . . . , xp) — independent variables

u = (u1, . . . , uq) — dependent variables

uαJ = ∂Juα — partial derivatives

F (x, u(n)) = F ( . . . xk . . . uαJ . . . ) — differential function

G — transformation group acting on the space of independent

and dependent variables

Variational Problems

I[u ] =∫

L(x, u(n)) dx — variational problem

L(x, u(n)) — Lagrangian

Variational derivative — Euler-Lagrange equations: E(L) = 0

components: Eα(L) =∑

J

(−D)J∂L

∂uαJ

DkF =∂F

∂xk+

∑

α,J

uαJ,k∂F

∂uαJ— total derivative of F with respect to xk

Invariant Variational Problems

According to Lie, any G–invariant variational problem can be

written in terms of the differential invariants:

I[u ] =∫

L(x, u(n)) dx =∫

P ( . . . DKIα . . . ) ω

I1, . . . , I" — fundamental differential invariants

D1, . . . ,Dp — invariant differential operators

DKIα — differentiated invariants

ω = ω1 ∧ · · · ∧ ωp — invariant volume form

If the variational problem is G-invariant, so

I[u ] =∫

L(x, u(n)) dx =∫

P ( . . . DKIα . . . ) ω

then its Euler–Lagrange equations admit G as a symmetrygroup, and hence can also be expressed in terms of the differ-ential invariants:

E(L) $ F ( . . . DKIα . . . ) = 0

Main Problem:

Construct F directly from P .

(P. Griffiths, I. Anderson )

Planar Euclidean group G = SE(2)

κ =uxx

(1 + u2x)3/2

— curvature (differential invariant)

ds =√

1 + u2x dx — arc length

D =d

ds=

1√

1 + u2x

d

dx— arc length derivative

Euclidean–invariant variational problem

I[u ] =∫

L(x, u(n)) dx =∫

P (κ,κs,κss, . . . ) ds

Euler-Lagrange equations

E(L) $ F (κ,κs,κss, . . . ) = 0

Euclidean Curve Examples

Minimal curves (geodesics):

I[u ] =∫

ds =∫ √

1 + u2x dx

E(L) = −κ = 0=⇒ straight lines

The Elastica (Euler):

I[u ] =∫

12 κ

2 ds =∫ u2xx dx

(1 + u2x)5/2

E(L) = κss +12 κ

3 = 0=⇒ elliptic functions

General Euclidean–invariant variational problem

I[u ] =∫

L(x, u(n)) dx =∫

P (κ,κs,κss, . . . ) ds

Invariantized Euler–Lagrange expression

E(P ) =∞∑

n=0

(−D)n∂P

∂κnD =

d

ds

Invariantized Hamiltonian

H(P ) =∑

i>j

κi−j (−D)j ∂P

∂κi− P

I[u ] =∫

L(x, u(n)) dx =∫

P (κ,κs,κss, . . . ) ds

Euclidean–invariant Euler-Lagrange formula

E(L) = (D2 + κ2) E(P ) + κH(P ) = 0

The Elastica: I[u ] =∫

12 κ

2 ds P = 12 κ2

E(P ) = κ H(P ) = −P = − 12 κ2

E(L) = (D2 + κ2) κ + κ (− 12 κ2 )

= κss +12 κ

3 = 0

Moving Frames

G — r-dimensional Lie group acting on M

Jn = Jn(M,p) — nth order jet bundle for

p-dimensional submanifolds N = {u = f(x)} ⊂M

z(n) = (x, u(n)) = ( . . . xi . . . uαJ . . . ) — coordinates on Jn

G acts on Jn by prolongation (chain rule)

Definition.

An nth order moving frame is a G-equivariant map

ρ = ρ(n) : V ⊂ Jn −→ G

Equivariance:

ρ(g(n) · z(n)) =

g · ρ(z(n)) left moving frame

ρ(z(n)) · g−1 right moving frame

Note: ρleft(z(n)) = ρright(z

(n))−1

Theorem. A moving frame exists in a neighborhoodof a point z(n) ∈ Jn if and only if G acts freelyand regularly near z(n).

• free — the only group element g ∈ G which fixes one point

z ∈M is the identity: g · z = z if and only if g = e.

• locally free — the orbits have the same dimension as G.

• regular — all orbits have the same dimension and intersect

sufficiently small coordinate charts only once

( )≈ irrational flow on the torus)

Geometric Construction

z

Oz

Normalization = choice of cross-section to the group orbits


z

Oz

K

k



z

Oz

K

k

g = ρleft(z)



z

Oz

K

k

g = ρright(z)


The Normalization Construction

1. Write out the explicit formulas for theprolonged group action:

w(n)(g, z(n)) = g(n) · z(n)

=⇒ Implicit differentiation

2. From the components of w(n), choose r = dim Gnormalization equations :

w1(g, z(n)) = c1 . . . wr(g, z

(n)) = cr

3. Solve the normalization equations for the group parametersg = (g1, . . . , gr):

g = ρ(z(n)) = ρ(x, u(n))

The solution is the right moving frame.

4. Invariantization: substitute the moving frame formulas

g = ρ(z(n)) = ρ(x, u(n))

for the group parameters into the un-normalized components ofw(n) to produce a complete system of functionally independentdifferential invariants:

I(n)(x, u(n)) = ι(z(n)) = w(n)(ρ(z(n)), z(n)))

Euclidean plane curves G = SE(2)

Assume the curve is (locally) a graph:

C = {u = f(x)}

Write out the group transformations

y = x cosφ− u sinφ + a

v = x cosφ + u sinφ + b

w = R z + c

Prolong to Jn via implicit differentiationy = x cosφ− u sinφ + a v = x cosφ + u sinφ + b

vy =sinφ + ux cosφ

cosφ− ux sinφvyy =

uxx(cosφ− ux sinφ)

3

vyyy =(cosφ − ux sinφ )uxxx − 3u

2xx sinφ

(cosφ − ux sinφ )5

...

Choose a cross-section, or, equivalently a set of r = dimG = 3normalization equations:

y = 0 v = 0 vy = 0

Solve the normalization equations for the group parameters:

φ = − tan−1 ux a = −x + uux√

1 + u2xb =

xux − u√1 + u2x

The result is the right moving frame ρ : J1 −→ SE(2)

Substitute into the moving frame formulas for the group pa-rameters into the remaining prolonged transformation formulaeto produce the basic differential invariants:

vyy +−→ κ =uxx

(1 + u2x)3/2

vyyy +−→dκ

ds=

(1 + u2x)uxxx − 3uxu2xx

(1 + u2x)3

vyyyy +−→d2κ

ds2+ 3κ3 = · · ·

Theorem. All differential invariants are functions of thederivatives of curvature with respect to arc length:

κdκ

ds

d2κ

ds2· · ·

The invariant differential operators and invariant differentialforms are also substituting the moving frame formulas forthe group parameters:

Invariant one-form — arc length

dy = (cosφ− ux sinφ) dx +−→ ds =√

1 + u2x dx

Invariant differential operator — arc length derivative

d

dy=

1

cosφ− ux sinφ

d

dx+−→

d

ds=

1√

1 + u2x

d

dx

Euclidean Curves

x

e1

e 2

Left moving frame ρ̃(x, u(1)) = ρ(x, u(1))−1

ã = x b̃ = u φ̃ = tan−1 ux

R =1

√1 + u2x

(1 −uxux 1

)

= ( t n ) ã =

(xu

)

InvariantizationThe process of replacing group parameters in transformation

rules by their moving frame formulae is known asinvariantization.

The invariantization I = ι(F ) is the unique invariant functionthat agrees with F on the cross-section: I | K = F | K.

Invariantization respects algebraic operations, and providesa canonical projection that maps objects to their invari-antized counterparts.

ι :

Functions −→ Invariants

Forms −→ Invariant Forms

Differential

Operators−→

Invariant Differential

Operators

Fundamental differential invariants = invariantized jet coordinates

Hi(x, u(n)) = ι(xi) IαK(x, u(l)) = ι(uαK)

The constant differential invariants, coming from the movingframe normalizations, are known as the phantom invariants .The remaining non-constant differential invariants are thebasic invariants and form a complete system of functionallyindependent differential invariants.

Invariantization of differential functions:

ι [ F ( . . . xi . . . uαJ . . . ) ] = F ( . . . Hi . . . IαJ . . . )

Replacement Theorem:

If J is a differential invariant, then ι(J) = J .

J( . . . xi . . . uαJ . . . ) = J( . . . Hi . . . IαJ . . . )

The Infinite Jet Bundle

Jet bundles

M = J0 ←− J1 ←− J2 ←− · · ·

Inverse limitJ∞ = lim

n→∞Jn

Local coordinates

z(∞) = (x, u(∞)) = ( . . . xi . . . uαJ . . . )

=⇒ Taylor series

Differential Forms

Coframe — basis for the cotangent space T∗J∞:

• Horizontal one-forms

dx1, . . . , dxp

• Contact (vertical) one-forms

θαJ = duαJ −

p∑

i=1

uαJ,i dxi

Intrinsic definition of contact form

θ | j∞N = 0 ⇐⇒ θ =∑

AαJ θαJ

The Variational Bicomplex=⇒ Dedecker, Vinogradov, Tsujishita, I. Anderson, . . .

Bigrading of the differential forms on J∞:

Ω∗ =M

r,sΩr,s

r = # horizontal forms

s = # contact forms

Vertical and Horizontal Differentials

d = dH + dV

dH : Ωr,s −→ Ωr+1,s

dV : Ωr,s −→ Ωr,s+1

Vertical and Horizontal Differentials

F (x, u(n)) — differential function

dH F =p∑

i=1

(DiF ) dxi — total differential (gradient)

dV F =∑

α,J

∂F

∂uαJθαJ — first variation

dH (dxi) = dV (dx

i) = 0,

dH (θαJ ) =

p∑

i=1

dxi ∧ θαJ,i dV (θαJ ) = 0

The Simplest Example

(x, u) ∈M = R2x — independent variable

u — dependent variable

Horizontal formdx

Contact (vertical) forms

θ = du− ux dx

θx = dux − uxx dx

θxx = duxx − uxxx dx

...

θ = du− ux dx, θx = dux − uxx dx, θxx = duxx − uxxx dx

Differential:

dF =∂F

∂xdx +

∂F

∂udu +

∂F

∂uxdux +

∂F

∂uxxduxx + · · ·

= (DxF ) dx +∂F

∂uθ +

∂F

∂uxθx +

∂F

∂uxxθxx + · · ·

= dH F + dV F

Total derivative:

DxF =∂F

∂x+

∂F

∂uux +

∂F

∂uxuxx +

∂F

∂uxxuxxx + · · ·

The Variational Bicomplex... ... ... ... ...

dV!

dV!

dV!

dV!

δ!

Ω0,3dH" Ω1,3

dH" · · ·dH" Ωp−1,3

dH" Ωp,3π

" F3

dV!

dV!

dV!

dV!

δ!

Ω0,2dH" Ω1,2

dH" · · ·dH" Ωp−1,2

dH" Ωp,2π

" F2

dV!

dV!

dV!

dV!

δ!

Ω0,1dH" Ω1,1

dH" · · ·dH" Ωp−1,1

dH" Ωp,1π

" F1

dV!

dV!

dV!

dV!

##

##$E

R→ Ω0,0dH" Ω1,0

dH" · · ·dH" Ωp−1,0

dH" Ωp,0

conservation laws Lagrangians PDEs (Euler–Lagrange) Helmholtz conditions

The Variational Derivative

E = π ◦ dV

dV — first variationπ — integration by parts = mod out by image of dH

Ωp,0 −→dV

Ωp,1 −→π

F1 = Ωp,1/ dH Ωp−1,1

λ = Ldx −→∑

α,J

∂L

∂uαJθαJ ∧ dx −→

q∑

α=1

Eα(L) θα ∧ dx

Variational

problem−→

First

variation−→

Euler–Lagrange

source form

The Simplest Example: (x, u) ∈M = R2

Lagrangian form: λ = L(x, u(n)) dx ∈ Ω1,0

First variation — vertical derivative:

dλ = dV λ = dV L ∧ dx

=

(∂L

∂uθ +

∂L

∂uxθx +

∂L

∂uxxθxx + · · ·

)

∧ dx ∈ Ω1,1

Integration by parts — compute modulo im dH :

dλ ∼ δλ =

(∂L

∂u−Dx

∂L

∂ux+ D2x

∂L

∂uxx− · · ·

)

θ ∧ dx ∈ F1

= E(L) θ ∧ dx

=⇒ Euler-Lagrange source form.

To analyze invariant variational problems,

invariant conservation laws, etc., we

apply the moving frame invariantization

process to the variational bicomplex:

The Invariant Variational Complex

=⇒ Joint work with Irina Kogan.

ι — invariantization associated with moving frame ρ.

• Fundamental differential invariants

Hi(x, u(n)) = ι(xi) IαK(x, u(n)) = ι(uαK)

• Invariant horizontal forms

+i = ι(dxi)

• Invariant contact forms

ϑαJ = ι(θαJ )

The Invariant “Quasi–Tricomplex”

Differential formsΩ∗ =

M

r,sΩ̂r,s

Differentiald = dH + dV + dW

dH : Ω̂r,s −→ Ω̂r+1,s

dV : Ω̂r,s −→ Ω̂r,s+1

dW : Ω̂r,s −→ Ω̂r−1,s+2

Key fact: Invariantization and differentiation do not commute:

d ι(Ω) )= ι(dΩ)

The Universal Recurrence Formula

d ι(Ω) = ι(dΩ) +r∑

k=1

νk ∧ ι [vk(Ω)]

v1, . . . ,vr — basis for g — infinitesimal generators

ν1, . . . , νr — invariantized dual Maurer–Cartan forms

ν = κ+ + C(ϑ)

κ — Maurer–Cartan invariantsC — Maurer–Cartan operator

=⇒ uniquely determined by the recurrence formulae for thephantom differential invariants

d ι(Ω) = ι(dΩ) +r∑

k=1

νk ∧ ι [vk(Ω)]

. . . All identities, commutation formulae, syzygies, etc.,among differential invariants and, more generally,the invariant variational bicomplex follow from thisuniversal formula by letting Ω range over the basicfunctions and differential forms!

. . . Moreover, determining the structure of the differentialinvariant algebra and invariant variational bicomplexrequires only linear differential algebra, and not anyexplicit formulas for the moving frame, the differentialinvariants, the invariant differential forms, or the grouptransformations!

Euclidean plane curves

Fundamental normalized differential invariants

ι(x) = H = 0

ι(u) = I0 = 0

ι(ux) = I1 = 0

phantom diff. invs.

ι(uxx) = I2 = κ ι(uxxx) = I3 = κs ι(uxxxx) = I4 = κss + 3κ3

In general:

ι( F (x, u, ux, uxx, uxxx, uxxxx, . . . )) = F (0, 0, 0,κ,κs,κss + 3κ3, . . . )

Invariant arc length form

dy = (cosφ− ux sinφ) dx − (sinφ) θ

+ = ι(dx) = ω + η

=√

1 + u2x dx +ux√

1 + u2xθ

=⇒ θ = du− ux dx

Invariant contact forms

ϑ = ι(θ) =θ

√1 + u2x

ϑ1 = ι(θx) =(1 + u2x) θx − uxuxxθ

(1 + u2x)2

Prolonged infinitesimal generators

v1 = ∂x, v2 = ∂u, v3 = −u ∂x + x ∂u + (1 + u2x) ∂ux + 3uxuxx ∂uxx + · · ·

Basic recurrence formula

dι(F ) = ι(dF ) + ι(v1(F )) ν1 + ι(v2(F )) ν

2 + ι(v3(F )) ν3

Use phantom invariants

0 = dH = ι(dx) + ι(v1(x)) ν1 + ι(v2(x)) ν

2 + ι(v3(x)) ν3 = + + ν1,

0 = dI0 = ι(du) + ι(v1(u)) ν1 + ι(v2(u)) ν

2 + ι(v3(u)) ν3 = ϑ + ν2,

0 = dI1 = ι(dux) + ι(v1(ux)) ν1 + ι(v2(ux)) ν

2 + ι(v3(ux)) ν3 = κ+ + ϑ1 + ν

3,

to solve for the Maurer–Cartan forms:

ν1 = −+, ν2 = −ϑ, ν3 = −κ+ − ϑ1.

ν1 = −+, ν2 = −ϑ, ν3 = −κ+ − ϑ1.

Maurer–Cartan invariants: κ = (−1, 0,−κ)

Recurrence formulae:

dκ = dι(uxx) = ι(duxx) + ι(v1(uxx)) ν1 + ι(v2(uxx)) ν

2 + ι(v3(uxx)) ν3

= ι(uxxx dx + θxx)− ι(3uxuxx) (κ+ + ϑ1) = I3+ + ϑ2.

Therefore, Dκ = κs = I3, dV κ = ϑ2 = (D2 + κ2)ϑ

where the final formula follows from the contact form recurrence formulae

dϑ = dι(θx) = + ∧ ϑ1, dϑ1 = dι(θ) = + ∧ (ϑ2 − κ2ϑ)− κϑ1 ∧ ϑ

which implyϑ1 = Dϑ, ϑ2 = Dϑ1 + κ

2 ϑ = (D2 + κ2)ϑ

Similarly,

d+ = ι(d2x) + ν1 ∧ ι(v1(dx)) + ν2 ∧ ι(v2(dx)) + ν

3 ∧ ι(v3(dx))

= (κ+ + ϑ1) ∧ ι(ux dx + θ) = κ+ ∧ ϑ + ϑ1 ∧ ϑ.

In particular,dV + = −κϑ ∧+

Key recurrence formulae:

dV κ = (D2 + κ2)ϑ dV + = −κ ϑ ∧+

Plane Curves

Invariant Lagrangian:

λ̃ = L(x, u(n)) dx = P (κ,κs, . . .) +

Euler–Lagrange form:dV λ̃ ∼ E(L)ϑ ∧+

Invariant Integration by Parts Formula

F dV (DH) ∧+ ∼ − (DF ) dV H ∧+ − (F · DH) dV +

dV λ̃ = dV P ∧+ + P dV +

=∑

n

∂P

∂κndV κn ∧+ + P dV +

∼ E(P ) dV κ ∧+ + H(P ) dV +

Vertical differentiation formulae

dV κ = A(ϑ) A — invariant variation operator

dV + = B(ϑ) ∧+ B — “Hamiltonian operator”

dV λ̃ ∼ E(P ) A(ϑ) ∧+ + H(P ) B(ϑ) ∧+

∼[A∗E(P )− B∗H(P )

]ϑ ∧+

Invariant Euler-Lagrange equation

A∗E(P )− B∗H(P ) = 0

Euclidean Plane Curves

dV κ = (D2 + κ2)ϑ

Invariant variation operator:

A = D2 + κ2 = A∗

dV + = −κ ϑ ∧+

Hamiltonian operator:

B = −κ = B∗

Euclidean–invariant Euler-Lagrange formula

E(L) = A∗E(P )− B∗H(P ) = (D2 + κ2) E(P ) + κH(P ).

Invariant Plane Curve Flows

G — Lie group acting on R2

C(t) — parametrized family of plane curves

G–invariant curve flow:

dC

dt= V = I t + J n

• I, J — differential invariants

• t — “unit tangent”

• n — “unit normal”

t, n — basis of the invariant vector fields dual to the invariantone-forms:

〈 t ;+ 〉 = 1, 〈n ;+ 〉 = 0,

〈 t ;ϑ 〉 = 0, 〈n ;ϑ 〉 = 1.

Ct = V = I t + J n

• The tangential component I t only affects the underlyingparametrization of the curve. Thus, we can set I to beanything we like without affecting the curve evolution.

• There are two principal choices of tangential component:

Normal Curve Flows

Ct = J n

Examples — Euclidean–invariant curve flows

• Ct = n — geometric optics or grassfire flow;

• Ct = κn — curve shortening flow;

• Ct = κ1/3 n — equi-affine invariant curve shortening flow:

Ct = nequi−affine ;

• Ct = κs n — modified Korteweg–deVries flow;

• Ct = κss n — thermal grooving of metals.

Intrinsic Curve Flows

Theorem. The curve flow generated by

v = I t + J n

preserves arc length if and only if

B(J) + D I = 0.

D — invariant arc length derivative

dV + = B(ϑ) ∧+

B — invariant Hamiltonian operator

Normal Evolution of Differential Invariants

Theorem. Under a normal flow Ct = J n,

∂κ

∂t= Aκ(J),

∂κs∂t

= Aκs(J).

Invariant variations:

dV κ = Aκ(ϑ), dV κs = Aκs(ϑ).

Aκ = A — invariant linearization operator of curvature;

Aκs = DAκ + κκs — invariant linearization operator of κs.

Euclidean–invariant Curve Evolution

Normal flow: Ct = J n

∂κ

∂t= Aκ(J) = (D

2 + κ2) J,

∂κs∂t

= Aκs(J) = (D3 + κ2D + 3κκs)J.

Warning : For non-intrinsic flows, ∂t and ∂s do not commute!

Grassfire flow: J = 1

∂κ

∂t= κ2,

∂κs∂t

= 3κκs, . . .

=⇒ caustics

Signature Curves

Definition. The signature curve S ⊂ R2 of a curve C ⊂ R2 is

parametrized by the two lowest order differential invariants

S =

{ (

κ ,dκ

ds

) }

⊂ R2

Equivalence and Signature Curves

Theorem. Two curves C and C are equivalent:

C = g · C

if and only if their signature curves are identical:

S = S

=⇒ object recognition

Euclidean Signature Evolution

Evolution of the Euclidean signature curve

κs = Φ(t,κ).

Grassfire flow:∂Φ

∂t= 3κΦ− κ2

∂Φ

∂κ.

Curve shortening flow:

∂Φ

∂t= Φ2Φκκ − κ

3Φκ + 4κ2Φ.

Modified Korteweg-deVries flow:

∂Φ

∂t= Φ3Φκκκ + 3Φ

2ΦκΦκκ + 3κΦ2.

Canine Left Ventricle Signature

Original Canine HeartMRI Image

Boundary of Left Ventricle

Smoothed Ventricle Signature

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Intrinsic Evolution of Differential Invariants

Theorem.

Under an arc-length preserving flow,

κt = R(J) where R = A− κsD−1B (∗)

In surprisingly many situations, (*) is a well-known integrable

evolution equation, and R is its recursion operator!

=⇒ Hasimoto

=⇒ Langer, Singer, Perline

=⇒ Maŕı–Beffa, Sanders, Wang

=⇒ Qu, Chou, Anco, and many more ...

Euclidean plane curves

G = SE(2) = SO(2) ! R2

dV κ = (D2 + κ2)ϑ, dV + = −κ ϑ ∧+

=⇒ A = D2 + κ2, B = −κ

R = A− κsD−1B = D2 + κ2 + κsD

−1 · κ

κt = R(κs) = κsss +32 κ

2κs

=⇒ modified Korteweg-deVries equation

Equi-affine plane curves

G = SA(2) = SL(2) ! R2

dV κ = A(ϑ), dV + = B(ϑ) ∧+

A = D4 + 53 κD2 + 53 κsD +

13 κss +

49 κ

2,

B = 13 D2 − 29 κ,

R = A− κsD−1B

= D4 + 53 κD2 + 43 κsD +

13 κss +

49 κ

2 + 29 κsD−1 · κ

κt = R(κs) = κ5s + 2κκss +43 κ

2s + +

59 κ

2 κs=⇒ Sawada–Kotera equation

Euclidean space curves

G = SE(3) = SO(3) ! R3

(dV κdV τ

)

= A

(ϑ1ϑ2

)

dV + = B

(ϑ1ϑ2

)

∧+

A =

D2s + (κ2 − τ2)

2τ

κD2s +

3κτs − 2κsτ

κ2Ds +

κτss − κsτs + 2κ3τ

κ2

−2τDs − τs

1

κD3s −

κsκ2

D2s +κ2 − τ2

κDs +

κsτ2 − 2κττsκ2

B = (κ 0 )

Recursion operator:

R = A−

(κsτs

)

D−1B

(κtτt

)

= R

(κsτs

)

=⇒ vortex filament flow

=⇒ nonlinear Schrödinger equation (Hasimoto)

Poisson Brackets

{ ·, · } : C∞(M, R)× C∞(M, R) −→ C∞(M, R)

Bilinear :

{ aF + bG, H } = a {F, H } + b {G, H }

{F, aG + bH } = a {F, G } + b {F, H }

Skew Symmetric:{F,H } = − {H, F }

Jacobi Identity :

{F, {G, H } } + {H, {F, G } } + {G, {H, F } } = 0

F,G, H ∈ C∞(M, R), a, b ∈ R.

Poisson Brackets

In coordinates z = (z1, . . . , zm),

{F, H } = ∇F TJ(z)∇H

where J(z)T = −J(z) is a skew symmetric matrix.

• The Jacobi identity imposes a system of quadraticallynonlinear partial differential equations on its entries Jij(z).

Symplectic Structure

A nondegenerate Poisson bracket

det J(z) )= 0

defines a symplectic structure, with symplectic 2–form

Ω = dzT J(z)−1 dz

Hamiltonian Flows

Given a Poisson structure, the Hamiltonian flow correspondingto H ∈ C∞(M, R) is the system of ordinary differentialequations

dz

dt= { z, H } = J(z)∇H

Canonical example:

z = (p, q) = (p1, . . . , pn, q1, . . . , qn) J(z) =

(O −II O

)

dp

dt= −

∂H

∂q

dq

dt=

∂H

∂p

Lie–Poisson Structure

g — r-dimensional Lie algebra

M = g∗ $ Rr — dual vector space

{F,H } = 〈 z ; [∇F (z),∇H(z) ] 〉

z ∈ g∗ F (z),H(z) ∈ C∞(g∗, R)

∇F (z) ∈ g [∇F (z),∇H(z) ] — Lie bracket in g

In coordinates: Jij(z) =r∑

k=1

ckijzk

ckij — structure constants

z = z1µ1 + · · · zrµ

r ∈ g∗

µ1, . . . , µr — Maurer–Cartan forms

Poisson Brackets on Curves

u(x) — parametrized curve in M = G/N

H[u ] =∫

H[u ] dx — Hamiltonian functional

Poisson bracket:

{F ,H} =∫ ( δF

δuPδH

δu

)

dx

P — skew-adjoint differential operator• Jacobi identity

Hamiltonian evolution

∂u

∂t= {u,H} = P

δH

δu

Korteweg–deVries Equation

∂u

∂t= uxxx + uux = P1

δH1δu

= P2δH2δu

P1 = Dx H1[u ] =∫

( 16 u3 − 12 u

2x ) dx

P2 = D3x +

23uDx +

13 ux H2[u ] =

∫( 12 u

2 ) dx

• Bi–Hamiltonian system with recursion operator

R = P2 · P−11 = D

2x +

23u +

13 ux D

−1x

. Magri’s Theorem establishes the integrability (under suitabletechnical hypotheses) of bi-Hamiltonian systems.

Deformed Lie–Poisson Structure

u(x) ∈ C∞(R, g∗) — curve in g∗

Poisson bracket:

{F ,H} =∫ ( δF

δuP

δH

δu

)

dx

Hamiltonian curve flow:

∂u

∂t= P

δH

δu= B Dx

δH

δu+ ad∗δH/δu(u)

B : g −→ g∗ — ad∗–invariant symmetric linear map

• If g is semi-simple, B is a multiple of the Killing form

Geometric Flows on Semi-Simple HomogeneousSpaces

=⇒ Joint work with Gloria Maŕı Beffa

From here on G is semi-simple and M = G/N .

We consider curve flows in M with fixed parametrization.

Let ρ : Jn → G be a left moving frame.

Maurer–Cartan invariants:

κ(u(n)) = ρ(u(n))−1Dxρ(u(n)) : Jn −→ g

Geometric curve flow∂u

∂t= J · n = J1n1 + · · · + Jmnm

Evolution of Maurer–Cartan invariants∂κ

∂t= A(J) = Q · C(J)

Poisson Reduction

Assuming g is semisimple, we use the Killing form to pull backthe Poisson structure on g∗ to g:

Q[v ] = B−1P[B v ], v ∈ C∞(R, g)

whereQ[v ] w = Dxw + [ v, w ]

We then replace v = κ(u(n)) using the Maurer–Cartan map topull back the Poisson flow to curves on M = G/N .

A remarkable fact is that the operator Q[κ ] is exactly theoperator appearing in the factorization of the Maurer–Cartanevolution

∂κ

∂t= A(J) = Q · C(J)

Moreover, the pulled back Maurer–Cartan forms have the form

ν = κ dx + C(ϑ)

Main questions:

• Does the pulled-back Poisson evolution of the Maurer–Cartaninvariants come from a geometric curve flow?

• Is the Maurer–Cartan flow Poisson?

Compatibility

Let κ1, . . . , κm be a generating system of differential invariants and assumethat they occur linearly in the Maurer–Cartan invariants:

κ = ϕ(κ) = A κ + b

Find R such thatδH

δκ[ϕ(κ) ] = B R

δH

δκThe reduced Poisson operator is

PR[κ ] = R∗P [B ϕ(κ) ]R

Compatibility condition M = G/N :

C(J) = RδH

δκ+ n for n ∈ n

Reduced Poisson Geometric Flow:

∂κ

∂t= PR

δH

δκ

Example

G = PSL(2) acting on M = R: u +−→αu + β

γ u + δ

Moving frame normalizations:

u→ 0, ux → 1, uxx → 0,

Generating differential invariant — the Schwarzian

κ = ι(uxxx) =2uxuxxx − 3u

2xx

u2x

Pulled back Maurer–Cartan forms

ν1 = −+ − ϑ, ν2 = −ϑx, ν3 = − 12 κ+ −

12 ϑxx −

12 κϑ.

Maurer–Cartan invariants and operator:

κ =

−10− 12 κ

C =

−1−D

− 12 D2 − 12 κ

Poisson operator on L g∗

P =

0 0 −10 12 0−1 0 0

Dx +

0 −L1 −2L2L1 0 −L12L2 L3 0

Pulled-back Poisson and Lie algebra operators

P =

0 −12 κ −Dx12 κ f2Dx −1−Dx 1 0

Q =

Dx −1 0κ Dx −20 12 κ Dx

Invariant linearization operator

A = Q · C =

00

− 12 (D3x + 2κDx + κx )

, R = −2 C =

2

2DxD2x + κ

Reduced Poisson operator = 2nd KdV structure:

PR = R∗ · P · R = −2 (D3x + 2κDx + κx)

Poisson flow:

κt = PRδh = −2 ( D3x + 2κDx + κx )

δh

δκ

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