MultiHamiltonian
structuresassociatedwith thespace ofclosed equicentroaﬃnecurvesMulti-Hamiltonian structures associated withthe space of closed equicentroaﬃne curves.

AtsushiFujioka
Contents
Introduction
Flows of equicentroaﬃnecurves
Hamiltonian
formalism
MultiHamiltonian
structures

Atsushi Fujioka..

.
Faculty of Engineering Science
Kansai University

December 7, 2012
Waseda University
Workshop on Curve ﬂows and integrable systems
Joint work with T. Kurose (Kwansei Gakuin University)
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Contents
MultiHamiltonian
structuresassociatedwith thespace ofclosed equicentroaﬃnecurves
AtsushiFujioka

1Introduction

2

Flows of equicentroaﬃne curves

3

Hamiltonian formalism

4

Multi-Hamiltonian structures..

Contents
Introduction
Flows of equicentroaﬃnecurves..

Hamiltonian
formalism
MultiHamiltonianstructures..

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Curve ﬂows
MultiHamiltonianstructuresassociatedwith thespace of
closed equicentroaﬃnecurves
AtsushiFujiokaContents
Introduction
Flows of equicentroaﬃnecurves◦ A curve ﬂow is a 1-parameter family of a curve.
◦ Geometric quantities vary under curve ﬂows.
◦ Soliton equations appear for special curve ﬂows.
◦ In some cases, Hamiltonian formalism can be applied.
◦ Also, it admits bi-Hamiltonian structure.

Hamiltonian
formalism
MultiHamiltonianstructures

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An example and the main result
MultiHamiltonian
structuresassociatedwith thespace of
closed equicentroaﬃnecurvesAtsushiFujiokaContents
Introduction
Flows of equicentroaﬃnecurvesHamiltonian
formalism
MultiHamiltonian
structures◦ Curve ﬂows associated with the KdV equation can beformulated as Hamiltonian system.
◦ 1995 U. Pinkall: Considered the space of closed
equicentroaﬃne curves to be an inﬁnite dimensional
symplectic manifold.
◦ The equicentroaﬃne curvature evolves according to the KdV
equation when the ﬂow is generated by a Hamiltonian
function given by the total equicentroaﬃne curvature.
◦ 2010 F-T. Kurose: Generalized Pinkall’s result to the case ofhigher KdV ﬂows.
◦ The above ﬂows also have bi-Hamiltonian structure.
◦ The main result: The level sets deﬁned by Hamiltonians forhigher KdV ﬂows have multi-Hamiltonian structures.
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Equicentroaﬃne curves
MultiHamiltonian
structuresassociatedwith thespace ofclosed equicentroaﬃnecurves
AtsushiFujioka.

Deﬁnition
I : an intervalγ : I → R2 : a plane curve
γ: an equicentroaﬃne curvem def
Any tangent line does not go through the origin.

.

Contents

γ : I → R2 : an equicentroaﬃne curve
Flows of equiChanging the variable, if necessary, may assume that the areal
centroaﬃnecurves
velocity is constant:
Hamiltonian()
formalismγ
Multidet= 1.
..
.
Hamiltonian
γ0Introduction

structures

We say the curve is parametrized by equicentroaﬃne arclength.
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Equicentroaﬃne curvature
MultiHamiltonian
structuresassociatedwith thespace ofclosed equicentroaﬃnecurves

γ : I → R2 : an equicentroaﬃne curve
Parametrized by equicentroaﬃne arclength:
()
γ
det
=1
γ0

AtsushiFujioka(Contents
Introduction
Flows of equicentroaﬃnecurves
Hamiltonian
formalism
MultiHamiltonian
structures

det⇓)

γ
γ 00
⇓

=0
(

γ 00 = − det

γ0γ 00)γ=: −κγ
κ: the equicentroaﬃne curvature
6 / 27Equicentroaﬃne curves with constant curvature
MultiHamiltonian
structuresassociatedwith thespace ofclosed equicentroaﬃnecurvesAtsushiFujiokaContents
Introduction
Flows of equicentroaﬃnecurvesHamiltonian
formalism
MultiHamiltonian
structures◦ If the equicentroaﬃne curvature is 0, the curve is a piece of aline which does not go through the origin.
◦ Ellipse:
a, b > 0(ss )γ(s) := a cos , b sin(s ∈ [0, 2πab])abab=⇒ s: an equicentroaﬃne arclength parameter, κ =◦ Hyperbola:a, b > 0(ss )γ(s) := a cosh , b sinhabab1a2 b 2(s ∈ R)

1=⇒ s: an equicentroaﬃne arclength parameter, κ = − 2 2a b
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The fundamental theorem of equicentroaﬃnecurvesMultiHamiltonian
structuresassociatedwith thespace ofclosed equicentroaﬃnecurvesEquicentroaﬃne transformations: Equiaﬃne transformations
ﬁxing the originx ∈ R2 7→ xA (A ∈ SL(2, R))

AtsushiFujiokaContents
Introduction
Flows of equicentroaﬃnecurvesHamiltonian
formalism
MultiHamiltonian
structures.

The fundamental theorem of equicentroaﬃne curves
I : an interval
κ:I →R=⇒ ∃ γ : I → R2 : an equicentroaﬃne curve withequicentroaﬃne curvature κ parametrized byequicentroaﬃne arclength
unique up to equicentroaﬃne transformations

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Flows of equicentroaﬃne curves
Consider a ﬂow of an equicentroaﬃne curve:

MultiHamiltonian
structuresassociatedwith thespace ofclosed equicentroaﬃnecurves

γ = γ(s, t) : I × J → R2
I , J: intervals
For each ﬁxed t ∈ J, γ(·, t) is an equicentroaﬃne curveparametrized by equicentroaﬃne arclength.

AtsushiFujioka

Proposition

Contents∃ α : I × J → R s.t.

Introduction
Flows of equicentroaﬃnecurvesHamiltonian
formalism
MultiHamiltonian
structures.

)()()
(γ0 1γ= γs−κ 0γs)()()s (αγ
γ− 12 αs=− 12 αss − κα 21 αsγsγs(∗)

t

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Proof of Proposition
MultiHamiltonian
structuresassociatedwith thespace of
closed equicentroaﬃnecurves.

ProofPutγt = βγ + αγs .
Then we have

AtsushiFujiokaγts = (βs − κα)γ + (β + αs )γs .

Contents
Introduction
Flows of equicentroaﬃnecurves
Hamiltonian
formalism
MultiHamiltonianstructures

Since γ(·, t) is parametrized by equicentroaﬃne arclength,()()γtγ
0 = det+ det
γsγst)()(γβγ + αγs+ det= det
γs(βs − κα)γ + (β + αs )γs= 2β + αs .

.
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The integrability condition
MultiHamiltonian
structuresassociatedwith thespace ofclosed equicentroaﬃnecurvesAtsushiFujiokaContents
Introduction

The integrability condition for the system of linear partial
diﬀerential equations (∗) is1
κt = αsss + 2καs + κs α.
2⇓1
κt = Ωαs , Ω = Ds2 + 2κ + κs Ds−1
2
Ω is the recursion operator of the KdV equation:

Flows of equicentroaﬃnecurvesHamiltonian
formalism
MultiHamiltonian
structures1
κt = κsss + 3κκs .
2
In particular, when
α = Ds−1 Ωn−1 κs(n ∈ N),

we have the nth KdV equation.
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The space of closed equicentroaﬃne curves
MultiHamiltonian
structuresassociatedwith thespace ofclosed equicentroaﬃnecurves
AtsushiFujioka

M: The space of closed equicentroaﬃne curves parametrizedby equicentroaﬃne arclength with enclosing area π()}{γ12=1M = γ : S → R detγs(S 1 = R/2πZ)

Flows of equicentroaﬃnecurvesγ∈M
From the system of linear partial diﬀerential equations (∗)}{11Tγ M = − αs γ + αγs α : S → R2Hamiltonianformalism

X , Y ∈ Tγ M

ContentsIntroduction

MultiHamiltonian
structures(∫ω0 (X , Y ) :=detS1XY)
ds12 / 27

A presymplectic form
MultiHamiltonian
structuresassociatedwith thespace of
closed equicentroaﬃnecurvesAtsushiFujiokaContents
Introduction
Flows of equicentroaﬃnecurvesHamiltonian
formalism
MultiHamiltonian
structures.

Proposition
ω0 deﬁnes a presymplectic form on M.

.

. . γ(·, t1 , t2 , t3 ) ∈ M: a 3-parameter family of an element of M.
ω0 : closedm∂∂∂ω0 (γt2 , γt3 ) +ω0 (γt3 , γt1 ) +ω0 (γt1 , γt2 ) = 0
∂t1
∂t2
∂t3
11X = − αs γ + αγs , Y = − βs γ + βγs (α, β : S 1 → R)
22⇓∫
ω0 (X , Y ) =αβs ds
S113 / 27

Closedness
MultiHamiltonian
structuresassociatedwith thespace of
closed equicentroaﬃnecurvesAtsushiFujiokaContents
Introduction
Flows of equicentroaﬃnecurvesHamiltonian
formalism
MultiHamiltonian
structuresγ(·, t1 , t2 , t3 ) ∈ M: a 3-parameter family of an element of M
Put1γti = − αis γ + αi γs (αi : S 1 → R, i = 1, 2, 3)
2
Since γti tj = γtj ti (i, j = 1, 2, 3), we have
αitj − αjti = αj αis − αi αjs .
On the other hand,∫∂∂ω0 (γt2 , γt3 ) =α2 α3s ds∂t1∂t1 S 1∫∫
=α2t1 α3s ds +α2 α3st1 ds
S1
S1
∫
∫
=α2t1 α3s ds −α2s α3t1 ds,
S1

S1

which vanishes by the cyclic sum.
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Hamiltonian functions
MultiHamiltonian
structuresassociatedwith thespace of
closed equicentroaﬃnecurvesThe nth KdV equation:κt = Ωn κs(κ : S 1 → R)

Introduction

has inﬁnite numbers of conserved quantities {Hm }m∈N
which can be represented as∫hm (κ, κs , κss , . . . )ds.
Hm =Flows of equicentroaﬃnecurvesFor example,

AtsushiFujiokaContents

Hamiltonian
formalism
MultiHamiltonian
structures

S1

1
1
1
h1 = κ, h2 = κ2 , h3 = κ3 − κ2s .
22
415 / 27

Hamiltonian formalism
MultiHamiltonian
structuresassociatedwith thespace ofclosed equicentroaﬃnecurvesAtsushiFujioka
Contents
Introduction
Flows of equicentroaﬃnecurvesHamiltonian
formalism
MultiHamiltonian
structures

Theorem (cf. U. Pinkall 1995 (n = 1), F-T. Kurose 2010).

n ∈ N: ﬁxed1
Xn := − (Ωn−1 κs )γ + (Ds−1 Ωn−1 κs )γs2(γ ∈ M)=⇒ Xn is a Hamiltonian vector ﬁeld for Hn with respect to ω0 :dHn = ω0 (Xn , · )In particular, Hn is a Hamiltonian function for the nth KdV
ﬂow:
γt = Xn .
.
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Another presymplectic form
MultiHamiltonian
structuresassociatedwith thespace ofclosed equicentroaﬃnecurvesAtsushiFujiokaContentsIntroduction
Flows of equicentroaﬃnecurvesHamiltonian
formalism
MultiHamiltonianstructures

X , Y ∈ Tγ M(∫detω1 (X , Y ) :=S1X(Ds2 + κ)Y)ds

1
1
X = − αs γ + αγs , Y = − βs γ + βγs (α, β : S 1 → R)22⇓∫
αΩβs ds
ω1 (X , Y ) =S1Theorem (F-T. Kurose)ω1 deﬁnes a presymplectic form on M.
Xn is a Hamiltonian vector ﬁeld for Hn+1 with respect to ω1 .
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Hamiltonian vector ﬁeld and Poisson structure
MultiHamiltonian
structuresassociatedwith thespace of
closed equicentroaﬃnecurves(M, ω): a symplectic manifold
H ∈ C ∞ (M)XH : a Hamiltonian vector ﬁeld
{ , }: the Poisson structure

AtsushiFujioka

Proposition
XH = { · , H}

Contents
Introduction

Proof
For f ∈ C ∞ (M),

Flows of equicentroaﬃnecurvesHamiltonian
formalism
MultiHamiltonian
structures.
..

XH f = df (XH )= ω(Xf , XH )= {f , H}.
.
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Bi-Hamiltonian structureMultiHamiltonian
structuresassociatedwith thespace ofclosed equicentroaﬃnecurves
AtsushiFujiokaContents
Introduction
Flows of equicentroaﬃnecurvesHamiltonian
formalism

Bi-Hamiltonian structure:◦ There exist two Poisson structures on a Poisson manifold.
{ , }1 , { , }2
◦ A Hamiltonian vector ﬁeld can be expressed in two ways.
{ · , H2 } 1 = { · , H1 } 2
ω0 and ω1 deﬁne bi-Hamiltonian structure on M.
dHn = ω0 (Xn , · ) = ω1 (Xn−1 , · )MultiHamiltonian
structures

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Magri’s theorem
MultiHamiltonian
structuresassociatedwith thespace ofclosed equicentroaﬃnecurvesAtsushiFujiokaContents
Introduction
Flows of equicentroaﬃnecurvesHamiltonian
formalism
MultiHamiltonian
structures

Theorem (F. Magri 1978).

M: a manifold with compatible Poisson structures i.e.
{ , }1 , { , }2 : Poisson structures{ , }1 + { , }2 : Poisson structure{ , }1 : non-degenerate (induced by a symplectic structure)∃ H , H ∈ C ∞ (M) s.t.
12

{ · , H2 } 1 = { · , H1 } 2=⇒ ∃ Hi ∈ C ∞ (M) (i ∈ N) s.t.
{ · , Hi+1 }1 = { · , Hi }2{Hi , Hj }1 = 0, {Hi , Hj }2 = 0(∀ i ∈ N)(∀ i, j ∈ N)

(1)
.
(2)
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Involutiveness
MultiHamiltonian
structuresassociatedwith thespace ofclosed equicentroaﬃnecurvesAtsushiFujioka

By involutiveness (2), {Hi }’s become ﬁrst integrals.
Proposition(2) can be deduced from (1).
.

Contents..

.

Proof
If i > j,{Hi , Hj }1 = {Hi−1 , Hj }2Introduction
Flows of equicentroaﬃnecurvesHamiltonianformalism
MultiHamiltonian
structures.

= {Hi−1 , Hj+1 }1 .
If 1 ≤ k < i,.

{Hi , Hj }1 = {Hi−k , Hj+k }1 = {Hi−k , Hj+k−1 }2 .
Devide into two cases that i − j is odd or even.
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The level sets of Hamiltonians
MultiHamiltonian
structuresassociatedwith thespace ofclosed equicentroaﬃnecurves
AtsushiFujiokaContentsm∈N
Cm := (c1 , . . . , cm ) ∈ Rm−1M(Cm ) := H1−1 (c1 ) ∩ · · · ∩ Hm(cm )Assume M(Cm ) 6= ∅.
γ ∈ M(Cm )Introduction
Flows of equicentroaﬃnecurvesHamiltonian
formalism
MultiHamiltonian
structures

Proposition

 1
Tγ M(Cm ) = − αs γ + αγs 21∫ α : Sk → RκΩ
αds=0sS1(k = 0, 1, 2, . . . , m − 1)

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Presymplectic forms on the level sets
MultiHamiltonian
structuresassociatedwith thespace of
closed equicentroaﬃnecurvesAtsushiFujiokaContents
Introduction
Flows of equicentroaﬃnecurvesHamiltonian
formalism
MultiHamiltonian
structures

Generalize ω0 and ω1 .
X , Y ∈ Tγ M
11X = − αs γ + αγs , Y = − βs γ + βγs22(α, β : S 1 → R)

k = 0, 1, 2, . . .
Assume Ωk αs and Ωk βs can be deﬁned.
∫αΩk βs ds
ωk (X , Y ) :=S1

Theorem (F-T. Kurose)ω0 , ω1 , . . . , ωm+1 deﬁne presymplectic forms on M(Cm ).
For each k = 0, 1, . . . , m + 1, Xn is a Hamiltonian vector ﬁeld
for Hn+k with respect to ωk .
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Key lemma
MultiHamiltonianstructuresassociatedwith thespace ofclosed equicentroaﬃnecurvesAtsushiFujiokaContents
Introduction
Flows of equicentroaﬃnecurvesHamiltonian
formalism
MultiHamiltonian
structuresp, q, r = 0, 1, 2, . . .
i = 1, 2, 3
Assume Ωp+1 αis , Ωq+1 αis and Ωr +1 αis can be deﬁned.
∫(Ωp α1s )(Ds−1 Ωq α2s )(Ωr α3s )ds
A(p, q, r ) :=1S∫+(Ωp α2s )(Ds−1 Ωq α3s )(Ωr α1s )ds
S1∫+(Ωp α3s )(Ds−1 Ωq α1s )(Ωr α2s )ds
S1

Lemma
A(p, q, r + 1) + A(q, r , p + 1) + A(r , p, q + 1)
− A(p + 1, q, r ) − A(q + 1, r , p) − A(r + 1, p, q) = 0
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Moment maps
MultiHamiltonian
structuresassociatedwith thespace ofclosed equicentroaﬃnecurvesAtsushiFujiokaContents
Introduction
Flows of equicentroaﬃnecurvesHamiltonian
formalism
MultiHamiltonian
structures

S 1 acts on M by parameter shift:M 3 γ 7→ γ(· + σ)(σ ∈ S 1 ),which is symplectic with respect to ω1 .
Moreover, the action is Hamiltonian and the moment map isgiven by H1 .
S 1 also acts on M(Cm ) by parameter shift, which is symplectic
with respect to ωm+1 .
Theorem (F-T. Kurose)
The moment map µm+1 for the S 1 -action on M(Cm ) withrespect to ωm+1 is given by( )∂µm+1 (γ)= Hm+1 (γ) (γ ∈ M(Cm )).
∂σ25 / 27

The Miura transformation
MultiHamiltonian
structuresassociatedwith thespace ofclosed equicentroaﬃnecurvesAtsushiFujiokaContents
Introduction
Flows of equicentroaﬃnecurvesHamiltonian
formalism
MultiHamiltonian
structures

The Miura transformation:√
κ=−11κ̂s + κ̂224If κ̂ is a solution to the mKdV equation:13κ̂t = κ̂sss + κ̂2 κ̂s ,24
κ is a solution to the KdV equation.
◦ Higher mKdV equations are associated with curve ﬂows in
the Euclidean plane.
◦ The Miura transformation can be deﬁned geometrically asmaps between complexiﬁcation of the set of closed curves.
◦ Multi-Hamiltonian structures are connected via the geometric
Miura transformation.
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MultiHamiltonianstructuresassociatedwith thespace ofclosed equicentroaﬃnecurves
AtsushiFujiokaThank you for your attention.

ContentsIntroduction
Flows of equicentroaﬃnecurves
Hamiltonian
formalism
MultiHamiltonian
structures

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