Hamiltonian S 1 -actions and integrable systems Sonja Hohloch (University of Antwerp, Belgium) September 14, 2021 Geometry, Dynamics and Mechanics Seminar
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!