Hamiltonian S1-actions

and

integrable systems

Sonja Hohloch(University of Antwerp, Belgium)

September 14, 2021

Geometry, Dynamics and Mechanics Seminar

Notation and conventions I

Definition: (M, ω) is a symplectic manifold if

I M smooth manifold,

I ω nondegenerate, closed 2-form on M.

dimM must be even.

(Standard) example: (M, ω) = (R2n,∑n

k=1 dpk ∧ dqk)where (q1, . . . , qn, p1, . . . , pn) =: (q, p) coordinates of R2n.

Darboux’s theorem:Any symplectic manifold looks locally like (R2n,

∑nk=1 dpk ∧ dqk).

locally all symplectic manifolds ‘look the same’, i.e., no ‘localsymplectic invariants’ like curvature in Riemannian geometry

Notation and conventions II

Definition: Let (M, ω) be symplectic and H : M → R smooth.

I ω(XH , ·) = dH Hamiltonian vector field

I z ′ = XH(z) Hamiltonian equation

In local coordinates (R2n, dp ∧ dq), we get

XH(q, p) =

(∂pH(q, p)

−∂qH(q, p)

)Hamiltonian vector field

and {q′ = ∂pH(q, p)

p′ = −∂qH(q, p)Hamiltonian equations

Special case: Hamiltonian S1-spaces:

Considering Hamiltonians with periodic flow with minimal period:

Classification by Karshon (1999) in dimension 4:

Let (M, ω) be a 4-dim compact sympl. manifold and let L : M → Rbe the momentum map of an effective Hamiltonian S1-action. Twosuch spaces are equivariantly symplectomorphic if and only if theirassociated labeled, directed graphs (next slide) are equal.

Notation: In what follows, Hamiltonians denoted by L often endup inducing an effective Hamiltonian S1-action.

Graph of S1-spaces:

(a)

2

2

2

(b) (c)

Graph:

I Vertex set: Fixed point = vertex, Fixed surface = fat vertex

I Edge set: Directed edges between vertices stand forZk -sphere, k ≥ 2, which are connected components of{x ∈ M | Stab(x) = Z/kZ}.

I Labels: value of moment map; fixed surface: volume & genus

Towards Liouville integrability: energy conservation

Energy conservation: H constant alongHamiltonian solution γ′ = XH(γ)

Ham. solutions stay within one level set (= ‘energy level’).

Conclusions:I Ham. sol. stay in ≤ dimM − 1 dimensional subsets.

I in dim = 2: regular Ham. sol. known up to parametrization.

Question: Can we confine solutions to even lower dim. subsets?

Liouville integrability:

Idea:Given H =: H1, look for more functions H2, H3, H4 . . . Hk

such that all Ham. sol. stay within each others’ level sets!

Then: each solution stays in intersection of level sets

Note:dim

(H−11 (r1) ∩ · · · ∩ H−1k (rk)

)≤ dimM − k

Let γHibe Hamiltonian solution of Hi and calculate:

(Hi ◦ γHj

)′= DHi .γ

′Hj

= DHi .XHj = ω(XHi ,XHj ) =: {Hi ,Hj}

Criterium:

γHjstays in level sets of Hi ⇔ {Hi ,Hj} = 0

Setting from now on:

(M, ω) 4-dimensional connected symplectic manifold.

Definitions and Notations:1) Φ = (L,H) : M → R2 is (the momentum map of) a

completely integrable Hamiltonian system ifI Ham. vector fields X L, XH almost everywhere lin. independentI L, H Poisson-commute: 0 = {L,H} := ω(X L,XH).

2) In particular, Hamiltonian flows commute: ϕLs ◦ ϕH

t = ϕHt ◦ ϕL

s

⇒ R2-action: R2 ×M → M, ((s, t), x) 7→ ϕLs ◦ ϕH

t (x).

Overview/Aim:

Toric systems

classified byDelzant 1988

(S1 × S1)-action

Semitoricsystems

classified byPelayo & Vu

Ngo.c 2009-2011

(S1 × R)-action

Hypersemitoricsystems

defined byHohloch & Palmer;

not yet classified

(S1 × R)-action

Hamiltonian S1-spaces satisfying:I g = 0

I at most twonon-free orbitsin eachL−1(`int)

Hamiltonian S1-spaces satisfying:I g = 0

I at most twonon-free,non-fixedorbits in eachL−1(`int)

All HamiltonianS1-spaces

classified byKarshon

1999

⊂ ⊂

⊂ ⊂

K 1999 HSS 2015K 1999 HSSS HP 2021

Essential ingredients: Singular points

(M, ω) 4-dimensional connected symplectic manifold.

Definitions and Notations:Φ = (L,H) : M → R2 completely integrable Hamiltonian system.

1) z singular point if rank DΦ|z < 2.

2) z nondegenerate fixed point ifI Hessians D2L|z and D2H|z linearly independent,I ∃ linear combination of ω−1D2L|z and ω−1D2H|z having

4 distinct eigenvalues.

Local normal form for nondegenerate singularities:

Theorem (Eliasson, Miranda & Zung. . . ):

Sympl. coordinates (x , ξ) = (x1, x2, ξ1, ξ2) and functions f1, f2 with{L, fk} = 0 = {H, fk} near nondegenerate singular point with:

1) Hyperbolic component: fk(x , ξ) = xkξk .

2) Elliptic component: fk(x , ξ) = 12(x2k + ξ2k).

3) Focus-Focus component (coming in pairs):{fk−1(x , ξ) = xk−1ξk − xkξk−1,

fk(x , ξ) = xk−1ξk−1 + xkξk .

4) Regular component: fk(x , ξ) = ξk .

Easiest integrable systems: Toric systems

Completely integrable systems Φ = (L,H) : M → R2 where theflows ϕL and ϕH are periodic of same period.

I R2-action becomes S1 × S1 =: T2-action

Example:

Φ : CP2 → R2, Φ([z0, z1, z2]) :=1

2

(|z1|2∑2k=0|zk |2

,|z2|2∑2k=0|zk |2

)

CP2 Φ−1(z)

z

⊃ell.-reg. ell.-ell.reg.

Φ

3Φ(CP2)

Singular points:

I elliptic-elliptic(rank = 0)

I elliptic-regular(rank = 1)

Classification of toric systems: Delzant’s Theorem

Delzant 1988: Constructive classification

Compact connected 2n-dim

symplectic manifolds (M, ω)

with effective Ham. Tn-action

/

equiv. sympl.

1:1−→

so-called

Delzant

polytopes

(M, ω,Tn,Φ) 7→ Φ(M).

Conclusion:I Toric manifolds determined by finite set of data.

I Toric manifolds = combinatorics

I Toric manifolds are a very special case of integrable systems.

Overview: Toric extension

Toric systems

classified byDelzant 1988

(S1 × S1)-action

Hamiltonian S1-spaces satisfying:I g = 0

I at most twonon-free orbitsin eachL−1(`int)

Karshon 1999Karshon 1999

Semitoric systems:

(M, ω) 4-dimensional connected symplectic manifold.

Definition (Pelayo & Vu Ngo.c)

A semitoric system is a completely integrable systemΦ = (L,H) : (M, ω)→ R2 such thatI L is proper,

I L induces an effective Hamiltonian S1-action,

I Φ admits only nondegenerate singularities,

I no hyperbolic components.

Conclusion: S1 × R-action; possible singularities:

I focus-focus (rank = 0)

I elliptic-elliptic (rank = 0)

I elliptic-regular (rank = 1)

Nondegenerate singularities (no hyp. components):

(M , ω) Φ−1(z)

z3

Φ

⊃

⊃R2

(L,H)

ell.-ell. ell.-reg.reg. focus-focus

Φ(M)Φ(M)

Semitoric systems:

The image of a (semitoric) momentum map Φ(M) is a ‘curvedpolygon’ which can be ‘straightened’ by some homeomorphismf (x , y) = (x , f (2)(x , y)) into polygon(s) with cuts:

H

c1 c1f

Lmin L

f (2)(L,H)

Lmin L

Semitoric systems: Classification

H

c1 c1f

Lmin L

f (2)(L,H)

Lmin L

Classification invariants (Pelayo & Vu Ngo.c):

(1) mFF , the number of focus-focus singularities.

(2) Taylor series invariant: Taylor series expansion of generatingfunction at focus-focus points.

(3) An equivalence class of generalized polygons.

(4) Height invariant: Height of focus-focus value in polygon.

(5) Twisting-index invariant: mFF numbers measuring ‘relativetwistedness’ near focus-focus singularities.

1st example: Coupled spin oscillators

(or Jaynes-Cummings, Gaudin model)

(S2 × R2, λωS2 ⊕ µωR2) with λ, µ > 0:

(L,H) : S2 × R2 → R2

EE FF

-2λ -λ λ 2λ 3λ 4λL

-λ

μ

λ

μ

H

L(x , y , z , u, v) = λ(z − 1) + µu2 + v2

2

H(x , y , z , u, v) =1

2(xu + yv)

2nd example: Coupled angular momenta

S2 × S2 with 0 < R1 < R2 and ω = −(R1ωS2 ⊕ R2ωS2),

t ∈ ]t−, t+[ ⊂ [0, 1].

(L,H) : S2 × S2 → R2

EE FF

EE

EE

-2R1 -2R1+2R2 2R2

-1

1/2

1/2

1

(figure for t = 12 )

L(x1, y1, z1, x2, y2, z2) = R1(z1 − 1) + R2(z2 + 1),

H(x1, y1, z1, x2, y2, z2) = (1− t)z1 + t(x1x2 + y1y2 + z1z2) + 2t − 1

3rd example: Family with 2 focus-focus singularities

Theorem (Hohloch & Palmer, 2018)S2 × S2 with R := (R1,R2), 0 < R1 < R2 andωR = R1ωS2 ⊕ R2ωS2 and t := (t1, t2, t3, t4) ∈ R4. Then{LR := R1z1 + R2z2,

H~t := t1z1 + t2z2 + t3(x1x2 + y1y2) + t4z1z2.

is semitoric with 2 focus-focus for certain parameters choices.

Theorem (Hohloch & Palmer, 2018)This system is semitoric for s1, s2 ∈ [0, 1], R1 = 1,R2 = 2 :t1 = s1(1− s2), t2 = s2(1− s1),t3 = (1− s1)(1− s2) + s1s2, t4 = (1− s1)(1− s2)− s1s2

Momentum map image of 2-parameter family:

Special parameter choices with s1, s2 ∈ [0, 1], R1 = 1,R2 = 2 :t1 = s1(1− s2), t2 = s2(1− s1),t3 = (1− s1)(1− s2) + s1s2, t4 = (1− s1)(1− s2)− s1s2

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

4th example: Le Floch & Palmer (2018) on W2(α, β):

Φs1,s2 = (L,Hs1,s2) : W2(α, β)→ R2

1 2 3-1

1

(s1,s2) = (0, 1)

1 2 3-1

1

(s1,s2) = (0.25, 1)

1 2 3-1

1

(s1,s2) = (0.5, 1)

1 2 3-1

1

(s1,s2) = (0.75, 1)

1 2 3-1

1

(s1,s2) = (1, 1)

1 2 3-1

1

(s1,s2) = (0, 0.75)

1 2 3-1

1

(s1,s2) = (0.25, 0.75)

1 2 3-1

1

(s1,s2) = (0.5, 0.75)

1 2 3-1

1

(s1,s2) = (0.75, 0.75)

1 2 3-1

1

(s1,s2) = (1, 0.75)

1 2 3-1

1

(s1,s2) = (0, 0.5)

1 2 3-1

1

(s1,s2) = (0.25, 0.5)

1 2 3-1

1

(s1,s2) = (0.5, 0.5)

1 2 3-1

1

(s1,s2) = (0.75, 0.5)

1 2 3-1

1

(s1,s2) = (1, 0.5)

1 2 3-1

1

(s1,s2) = (0, 0.25)

1 2 3-1

1

(s1,s2) = (0.25, 0.25)

1 2 3-1

1

(s1,s2) = (0.5, 0.25)

1 2 3-1

1

(s1,s2) = (0.75, 0.25)

1 2 3-1

1

(s1,s2) = (1, 0.25)

1 2 3-1

1

(s1,s2) = (0, 0)

1 2 3-1

1

(s1,s2) = (0.25, 0)

1 2 3-1

1

(s1,s2) = (0.5, 0)

1 2 3-1

1

(s1,s2) = (0.75, 0)

1 2 3-1

1

(s1,s2) = (1, 0)

(here: α = 1 and β = 1)

5th example: De Meulenaere & Hohloch (2021):

1-parameter family with simultaneous moving of singular points:

8 EE 8 EE 4 EE + 4 FF 4 EE + 4 FF

single pinched double pinched

t = 0 0 < t < t− t− < t < 12 t = 1

2

Overview classification progress I

Some of the symplectic invariants had been calculated for certainparameter values of

I spherical pendulum

I coupled spin oscillator (or Jaynes-Cummings, Gaudin model)

I coupled angular momenta

in various works by

Dullin, Pelayo, Vu Ngoc, Le Floch, Babelon and others...

Open problem: Twisting index; higher order terms of Taylor seriesinvariant; continuous parameters (for all invariants)...

Overview classification progress II:

Alonso & Dullin & Hohloch (2017, 2018):

I Taylor series and twisting index for coupled spin oscillator. Pelayo & Vu Ngo.c’s classification completed!

I Taylor series, height invariant, twisting index, (and polygoninvariant) for coupled angular momenta. Pelayo & Vu Ngo.c’s classification completed!

I Also calculated for more general parameter values of coupledangular momenta than the here formulated system...

I Limit behaviour of Taylor series of coupled angular momentafor t → t±: it blows up!

Overview classification progress III:

Le Floch & Palmer’s Hirzebruch systems:

I number of FF and polygon invariant for all parameter values(Le Floch & Palmer 2018)

I height invariant for selected parameter values(Le Floch & Palmer 2018)

still unknown:

I height invariant for all parameter values

I Taylor series invariant

I twisting index

Overview classification progress IV:

Hohloch & Palmer’s 2-FF-system:

I number of FF and polygon invariant(Hohloch & Palmer 2018)

I height invariant for selected parameter values(Le Floch & Palmer 2018/19)

I height invariant for 1-parameter subfamily(Alonso & Hohloch 2021)

I Taylor series invariant and Twisting index(Alonso & Hohloch & Palmer, ongoing)

De Meulenaere & Hohloch’s octagon-system:

I number of FF(De Meulenaere & Hohloch 2019)

I ... ?

Overview: toric and semitoric extensions

Toric systems

classified byDelzant 1988

(S1 × S1)-action

Semitoricsystems

classified byPelayo & Vu

Ngo.c 2009-2011

(S1 × R)-action

HSS 2015 =Hohloch & S.

Sabatini & D. Sepe

HSSS (ongoing)

= Hohloch & S.

Sabatini & D. Sepe

& M. Symington

Hamiltonian S1-spaces satisfying:I g = 0

I at most twonon-free orbitsin eachL−1(`int)

Hamiltonian S1-spaces satisfying:I g = 0

I at most twonon-free,non-fixed orbitsin eachL−1(`int)

⊂

⊂

K 1999 HSS 2015K 1999 HSSS

Hypersemitoric systems I:

Definition (Hohloch & Palmer 2021)

(M, ω) 4-dim, compact, connected symplectic manifold.A completely integrable system Φ = (L,H) : M → R2 ishypersemitoric if

I L induces an effective S1-action,

I all singularities of Φ are nondegenerate except for maybefinitely many parabolic degenerate points.

Possible nondegenerate singularities:

Rank 0: elliptic-elliptic, elliptic-hyperbolic, focus-focus.Rank 1: elliptic-regular, hyperbolic-regular.

Note: The periodic flow of L prevents the existence ofhyperbolic-hyperbolic fixed points.

Hyperbolic-regular and parabolic degenerate points:

Flap Swallowtail

Definition of parabolic degenerate points(M, ω,Φ) integrable system, p ∈ M a singular with df1(p) 6= 0where (f1, f2) = g ◦ Φ for some local diffeormorphism g of R2 inneighborhood of Φ(p). Define

f2 := f2,p := (f2)|f −11 (f1(p))

: f −11 (f1(p))→ R.

p is a parabolic degenerate singular point if:

I p is a critical point of f2,

I rank(d2f2(p)) = 1,

I there exists v ∈ ker(d2f2(p)) such that

v3(f2) :=d3

dt3f2(γ(t))|t=0

is nonzero, where γ : ]− ε, ε[ → f −11 (f1(p)) is a curvesatisfying γ(0) = p and γ(0) = v .

I rank(d2(f2 − kf1)(p)) = 3, where k ∈ R is determined bydf2(p) = kdf1(p).

Hypersemitoric systems II:

Theorem (Hohloch & Palmer 2021)

Let (M, ω) be a 4-dim, compact, connected symplectic manifold.Then for any Hamiltonian L that induces an effective S1-actionthere exists a smooth H : M → R such that Φ := (L,H) : M → R2

is a hypersemitoric system.

RemarkI Some L force the existence of degenerate points in any

extension. But one can choose what type of degenerate pointsone wants to admit (we opted for parabolic points).

I Hypersemitoric systems are the ‘nicest and easiest’ class ofintegrable systems to which an arbitrary L that induces aneffective S1-action can be extended.

Creating/deleting fixed points (ideas of HSSS):

(a) The Delzant polygon ofBl4(CP2).

×

(b) A semitoric polygon ofBl5(CP2).

2

2

(c) The Karshon graph of theHamiltonian S1-space induced bythe toric system.

2

2

(d) The Karshon graph of theHamiltonian S1-space induced bythe semitoric system.

Creating/deleting inner edges (HP):

Karshon graphKarshon graphKarshon graph

Dullin

Pelayo

blowup

2

2 2

2

2

2

2

Overview of Results:

Toric systems

classified byDelzant 1988

(S1 × S1)-action

Semitoricsystems

classified byPelayo & Vu

Ngo.c 2009-2011

(S1 × R)-action

Hypersemitoricsystems

defined byHohloch & Palmer;

not yet classified

(S1 × R)-action

Hamiltonian S1-spaces satisfying:I g = 0

I at most twonon-free orbitsin eachL−1(`int)

Hamiltonian S1-spaces satisfying:I g = 0

I at most twonon-free,non-fixedorbits in eachL−1(`int)

All HamiltonianS1-spaces

classified byKarshon

1999

⊂ ⊂

⊂ ⊂

K 1999 HSS 2015K 1999 HSSS HP 2021

Thank you for your attention!