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Digital Object Identiﬁer (DOI) https://doi.org/10.1007/s00220-019-03578-2

Commun. Math. Phys. 375, 1373–1392 (2020) Communications in

Mathematical

Physics

Hamiltonian Monodromy and Morse Theory

N. Martynchuk1,2,H.W.Broer

1, K. Efstathiou1

1Bernoulli Institute, University of Groningen, P.O. Box 407, 9700 AK Groningen, The Netherlands

2Present address: Department Mathematik, FAU Erlangen-Nürnberg, Cauerstr. 11, 91058 Erlangen,

Germany. E-mail: nmartynchuk@gmail.com

Received: 1 January 2019 / Accepted: 31 July 2019

Published online: 1 October 2019 – © The Author(s) 2019

Abstract: We show that Hamiltonian monodromy of an integrable two degrees of free-

dom system with a global circle action can be computed by applying Morse theory to the

Hamiltonian of the system. Our proof is based on Takens’s index theorem, which spec-

iﬁes how the energy-hChern number changes when hpasses a non-degenerate critical

value, and a choice of admissible cycles in Fomenko–Zieschang theory. Connections of

our result to some of the existing approaches to monodromy are discussed.

1. Introduction

Questions related to the geometry and dynamics of ﬁnite-dimensional integrable Hamil-

tonian systems [2,10,15] permeate modern mathematics, physics, and chemistry. They

are important to such disparate ﬁelds as celestial and galactic dynamics [8], persistence

and stability of invariant tori (Kolmogorov–Arnold–Moser and Nekhoroshev theories)

[1,12,35,47,53], quantum spectra of atoms and molecules [14,16,52,59], and the SYZ

conjecture in mirror symmetry [56].

At the most fundamental level, a local understanding of such systems is provided by

the Arnol’d–Liouville theorem [2,3,37,46]. This theorem states that integrable systems

are generically foliated by tori, given by the compact and regular joint level sets of the

integrals of motion, and that such foliations are always locally trivial (in the symplectic

sense). A closely related consequence of the Arnol’d–Liouville theorem, is the local

existence of the action coordinates given by the formula

Ii=αi

pdq,

where αi,i=1,...,n,are independent homology cycles on a given torus Tnof the

foliation.

Passing from the local to the global description of integrable Hamiltonian systems,

naturally leads to questions on the geometry of the foliation of the phase space by

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1374 N. Martynchuk, H. W. Broer, K. Efstathiou

Arnol’d–Liouville tori. For instance, the question of whether the bundles formed by

Arnol’d–Liouville tori come from a Hamiltonian torus action, is closely connected to the

existence of global action coordinates and Hamiltonian monodromy [20]. In the present

work, we shall review old and discuss new ideas related to this classical invariant.

Monodromy was introduced by Duistermaat in [20] and it concerns a certain ‘holon-

omy’ effect that appears when one tries to construct global action coordinates for a given

integrable Hamiltonian system. If the homology cycles αiappearing in the deﬁnition of

the actions Iicannot be globally deﬁned along a certain closed path in phase space, then

the monodromy is non-trivial; in particular, the system has no global action coordinates

and does not admit a Hamiltonian torus action of maximal dimension (the system is not

toric).

Non-trivial Hamiltonian monodromy was found in various integrable systems. The

list of examples contains among others the (quadratic) spherical pendulum [7,15,20,27],

the Lagrange top [17], the Hamiltonian Hopf bifurcation [21], the champagne bottle [6],

the Jaynes–Cummings model [23,33,49], the Euler two-center and the Kepler prob-

lems [26,39,61]. The concept of monodromy has also been extended to near-integrable

systems [11,13,51].

In the context of monodromy and its generalizations, it is natural to ask how one

can compute this invariant for a given class of integrable Hamiltonian systems. Since

Duistermaat’s work [20], a number of different approaches to this problem, ranging from

the residue calculus to algebraic and symplectic geometry, have been developed. The

very ﬁrst topological argument that allows one to detect non-trivial monodromy in the

spherical pendulum has been given by Richard Cushman. Speciﬁcally, he observed that,

for this system, the energy hyper-surfaces H−1(h)for large values of the energy hare

not diffeomorphic to the energy hyper-surfaces near the minimum where the pendulum

is at rest. This property is incompatible with the triviality of monodromy; see [20] and

Sect. 3for more details. This argument demonstrates that the monodromy in the spherical

pendulum is non-trivial, but does not compute it.

Cushman’s argument had been sleeping for many years until Floris Takens [57]

proposed the idea of using Chern numbers of energy hyper-surfaces and Morse theory

for the computation of monodromy. More speciﬁcally, he observed that in integrable

systems with a Hamiltonian circle action (such as the spherical pendulum), the Chern

number of energy hyper-surfaces changes when the energy passes a critical value of

the Hamiltonian function. The main purpose of the present paper is to explain Takens’s

theorem and to show that it allows one to compute monodromy in integrable systems

with a circle action.

We note that the present work is closely related to the works [30,40], which demon-

strate how one can compute monodromy by focusing on the circle action and without

using Morse theory. However, the idea of computing monodromy through energy hyper-

surfaces and their Chern numbers can also be applied when we do not have a detailed

knowledge of the singularities of the system; see Remark 8. In particular, it can be ap-

plied to the case when we do not have any information about the ﬁxed points of the circle

action. We note that the behaviour of the circle action near the ﬁxed points is important

for the theory developed in the works [30,40].

The paper is organized as follows. In Sect. 2we discuss Takens’s idea following [57].

In particular, we state and prove Takens’s index theorem, which is central to the present

work. In Sect. 3we show how this theorem can be applied to the context of monodromy.

We discuss in detail two examples and make a connection to the Duistermaat–Heckman

theorem [22]. In Sect. 4we revisit the symmetry approach to monodromy presented in

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Hamiltonian Monodromy and Morse Theory 1375

the works [30,40], and link it to the rotation number [15]. The paper is concluded with

a discussion in Sect. 5. Background material on Hamiltonian monodromy and Chern

classes is presented in the Appendix.

2. Takens’s Index Theorem

We consider an oriented 4-manifold Mand a smooth Morse function Hon this manifold.

We recall that His called a Morse function if for any critical (= singular) point xof H,

the Hessian

∂2H

∂xi∂xj

(x)

is non-degenerate. We shall assume that His a proper1function and that it is invariant

under a smooth circle action G:M×S1→Mthat is free outside the critical points of

H. Note that the critical points of Hare the ﬁxed points of the circle action.

Remark 1 (Context of integrable Hamiltonian systems). In the context of integrable

systems, the function His given by the Hamiltonian of the system or another ﬁrst

integral, while the circle action comes from the (rotational) symmetry. For instance, in

the spherical pendulum [15,20], which is a typical example of a system with monodromy,

one can take the function Hto be the Hamiltonian of the system; the circle action is

given by the component of the angular momentum along the gravitational axis. We shall

discuss this example in detail later on. In the Jaynes–Cummings model [23,33,49], one

can take the function Hto be the integral that generates the circle action, but one can

not take Hto be the Hamiltonian of the system since the latter function is not proper.

For any regular level Hh={x∈M|H(x)=h},the circle action gives rise to the

circle bundle

ρh:Hh→Bh=Hh/S1.

By deﬁnition, the ﬁbers ρ−1

h(b)of this bundle ρhare the orbits of the circle action. The

question that was addressed by Takens is how the Chern number (also known as the

Euler number since it generalizes the Euler characteristic) of this bundle changes as h

passes a critical value of H. Before stating his result we shall make a few remarks on

the Chern number and the circle action.

First, we note that the manifolds Hhand Bhare compact and admit an induced

orientation. Assume, for simplicity, that Bh(and hence Hh) are connected. Since the

base manifold Bhis 2-dimensional, the (principal) circle bundle ρh:Hh→Bhhas an

‘almost global’ section

s:Bh→ρ−1

h(Bh)

that is not deﬁned at most in one point b∈bh.Let αbe a (small) loop that encircles this

point.

1Preimages of compact sets are compact.

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1376 N. Martynchuk, H. W. Broer, K. Efstathiou

Deﬁnition 1. The Chern number c(h)of the principal bundle

ρh:Hh→Bh

can be deﬁned as the winding number of s(α) along the orbit ρ−1

h(b). In other words,

c(h)is the degree of the map

S1=α→s(α) →ρ−1

h(b)=S1,

where the map s(α) →ρ−1

h(b)is induced by a retraction of a tubular neighbourhood of

ρ−1

h(b)onto ρ−1

h(b).

Remark 2. We note that the Chern number c(h)is a topological invariant of the bundle

ρh:Hh→Bhwhich does not depend on the speciﬁc choice of the section sand the

loop α; for details see [31,45,50].

Now, consider a singular point Pof H. Observe that this point is ﬁxed under the

circle action. From the slice theorem [4, Theorem I.2.1] (see also [9]) it follows that in

a small equivariant neighbourhood of this point the action can be linearized. Thus, in

appropriate complex coordinates (z,w) ∈C2the action can be written as

(z,w) → (eimtz,eintw), t∈S1,

for some integers mand n. By our assumption, the circle action is free outside the

(isolated) critical points of the Morse function H. Hence, near each such critical point

the action can be written as

(z,w) → (e±itz,eit w), t∈S1,

in appropriate complex coordinates (z,w) ∈C2. The two cases can be mapped to each

other through an orientation-reversing coordinate change.

Deﬁnition 2. A singular point Pis called positive if the local circle action is given by

(z,w) → (e−itz,eit w) and negative if the action is given by (z,w) → (eitz,eit w) in

a coordinate chart having the positive orientation with respect to the orientation of M.

Remark 3. The Hopf ﬁbration is deﬁned by the circle action (z,w) → (eit z,eit w) on

the sphere

S3={(z,w)∈C2|1=|z|2+|w|2}.

The circle action (z,w) → (e−itz,eit w) deﬁnes the anti-Hopf ﬁbration on S3[58]. If

the orientation is ﬁxed, these two ﬁbrations are different.

Lemma 1. The Chern number of the Hopf ﬁbration is equal to −1, while for the anti-

Hopf ﬁbration it is equal to 1.

Proof. See Appendix B.

Theorem 1 (Takens’s index theorem [57]). Let H be a proper Morse function on an

oriented 4-manifold. Assume that H is invariant under a circle action that is free outside

the critical points. Let hcbe a critical value of H containing exactly one critical point.

Then the Chern numbers of the nearby levels satisfy

c(hc+ε) =c(hc−ε) ±1.

Here the sign is plus if the circle action deﬁnes the anti-Hopf ﬁbration near the critical

point and minus for the Hopf ﬁbration.

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Hamiltonian Monodromy and Morse Theory 1377

Proof. The main idea is to apply Morse theory to the function H. The role of Euler

characteristic in standard Morse theory will be played by the Chern number. We note

that the Chern number, just like the Euler characteristic, is additive.

From Morse theory [44], we have that the manifold H−1(−∞,hc+ε]can be obtained

from the manifold H−1(−∞,hc−ε]by attaching a handle Dλ×D4−λ, where λis

the index of the critical point on the level H−1(hc). More speciﬁcally, for a suitable

neighbourhood Dλ×D4−λ⊂Mof the critical point (with Dmstanding for an m-

dimensional ball), H−1(−∞,hc+ε]deformation retracts onto the set

X=H−1(−∞,hc−ε]∪Dλ×D4−λ

and, moreover,

H−1(−∞,hc+ε]X=H−1(−∞,hc−ε]∪Dλ×D4−λ(1)

up to a diffeomorphism. We note that by the construction, the intersection of the handle

Dλ×D4−λwith H−1(−∞,hc−ε]is the subset Sλ−1×D4−λ⊂H−1(hc−ε);see

[44]. For simplicity, we shall assume that the handle is disjoint from H−1(hc+ε).By

taking the boundary in Eq. (1), we get that

H−1(hc+ε) ∂X=(H−1(hc−ε)\Sλ−1×D4−λ)∪Dλ×S4−λ−1.(2)

Here the union (Dλ×S4−λ−1)∪(Sλ−1×D4−λ)is the boundary S3=∂(Dλ×D4−λ)

of the handle.

Since we assumed the existence of a global circle action on M, we can choose the

handle and its boundary S3to be invariant with respect to this action [62]. This will

allow us to relate the Chern numbers of H−1(hc+ε) and H−1(hc−ε) using Eq. (2).

Speciﬁcally, due to the invariance under the circle action, the sphere S3has a well-deﬁned

Chern number. Moreover, since the action is assumed to be free outside the critical points

of H, this Chern number c(S3)=±1, depending on whether the circle action deﬁnes

the anti-Hopf or the Hopf ﬁbration on S3; see Lemma 1.FromEq.(2) and the additive

property of the Chern number, we get

c(∂ X)=c(hc−ε) +c(S3)=c(hc−ε) ±1.

It is left to show that c(hc+ε) =c(∂ X)(we note that even though we know that

H−1(hc+ε) and ∂Xare diffeomorphic, we cannot yet conclude that they have the same

Chern numbers).

Let the subset Y⊂Mbe deﬁned as the closure of the set

H−1[hc−ε, hc+ε]\Dλ×D4−λ.

We observe that Yis a compact submanifold of Mand that ∂Y=∂X∪H−1(hc+ε),

that is, Yis a cobordism in Mbetween ∂Xand H−1(hc+ε). By the construction, ∂Y

is invariant under the circle action and there are no critical points of Hin Y. It follows

that the Chern number c(∂Y)=0. Indeed, one can apply Stokes’s theorem to the Chern

class of ρ:Y→Y/S1, where ρis the reduction map; see Appendix B. This concludes

the proof of the theorem.

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1378 N. Martynchuk, H. W. Broer, K. Efstathiou

Remark 4. We note that (an analogue of) Theorem 1holds also when the Hamiltonian

function Hhas k>1 isolated critical points on a critical level. In this case

c(hc+ε) =c(hc−ε) +

k

i=1

sk,

where sk=±1 corresponds to the kth critical point.

Remark 5. By a continuity argument, the (integer) Chern number is locally constant.

This means that if [a,b]does not contain critical values of H, then c(h)is the same

for all the values h∈[a,b]. On the other hand, by Theorem 1, the Chern number c(h)

changes when hpasses a critical value which corresponds to a single critical point.

3. Morse Theory Approach to Monodromy

The goal of the present section is to show how Takens’s index theorem can be used

to compute Hamiltonian monodromy. First, we demonstrate our method on a famous

example of a system with non-trivial monodromy: the spherical pendulum. Then, we

give a new proof of the geometric monodromy theorem along similar lines. We also show

that the jump in the energy level Chern number manifests non-triviality of Hamiltonian

monodromy in the general case. This section is concluded with studying Hamiltonian

monodromy in an example of an integrable system with two focus–focus points.

3.1. Spherical pendulum. The spherical pendulum describes the motion of a particle

moving on the unit sphere

S2={(x,y,z)∈R3:x2+y2+z2=1}

in the linear gravitational potential V(x,y,z)=z.The corresponding Hamiltonian

system is given by

(T∗S2,Ω|T∗S2,H|T∗S2), where H=1

2(p2

x+p2

y+p2

z)+V(x,y,z)

is the total energy of the pendulum and Ωis the standard symplectic structure. We observe

that the function J=xpy−ypx(the component of the total angular momentum about

the z-axis) is conserved. It follows that the system is Liouville integrable. The bifurcation

diagram of the energy-momentum map

F=(H,J):T∗S2→R2,

that is, the set of the critical values of this map, is shown in Fig. 1.

From the bifurcation diagram we see that the set R⊂image(F)of the regular values

of F(the shaded area in Fig. 2) is an open subset of R2with one puncture. Topologically,

Ris an annulus and hence π1(R,f0)=Zfor any f0∈R. We note that the puncture (the

black dot in Fig. 1) corresponds to an isolated singularity; speciﬁcally, to the unstable

equilibrium of the pendulum.

Consider the closed path γaround the puncture that is shown in Fig. 1. Since J

generates a Hamiltonian circle action on T∗S2, any orbit of this action on F−1(γ (0))

can be transported along γ.Let(a,b)be a basis of H1(F−1(γ (0))), where bis given by

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Hamiltonian Monodromy and Morse Theory 1379

Fig. 1. Bifurcation diagram for the spherical pendulum and the loop γaround the focus–focus singularity

the homology class of such an orbit. Then the corresponding Hamiltonian monodromy

matrix along γis given by

Mγ=1mγ

01

for some integer mγ.Itwasshownin[20] that mγ=1 (in particular, global action

coordinates do not exist in this case). Below we shall show how this result follows from

Theorem 1.

First we recall the following argument due to Cushman, which shows that the mon-

odromy along the loop γis non-trivial; the argument appeared in [20].

Cushman’s argument. First observe that the points

Pmin ={p=0,z=−1}and Pc={p=0,z=1}

are the only critical points of H. The corresponding critical values are hmin =−1 and

hc=1, respectively. The point Pmin is the global and non-degenerate minimum of Hon

T∗S2. From the Morse lemma, we have that H−1(1−ε), ε ∈(0,2), is diffeomorphic

to the 3-sphere S3. On the other hand, H−1(1+ε) is diffeomorphic to the unit cotangent

bundle T∗

1S2. It follows that the monodromy index mγ= 0. Indeed, the energy levels

H−1(1+ε) and H−1(1−ε) are isotopic, respectively, to F−1(γ1)and F−1(γ2), where

γ1and γ2are the curves shown in Fig. 2.Ifmγ=0, then the preimages F−1(γ1)and

F−1(γ2)would be homeomorphic, which is not the case.

Using Takens’s index theorem 1, we shall now make one step further and compute

the monodromy index mγ. By Takens’s index theorem, the energy-level Chern numbers

are related via

c(1+ε) =c(1−ε) +1

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1380 N. Martynchuk, H. W. Broer, K. Efstathiou

Fig. 2. Bifurcation diagram for the spherical pendulum, the energy levels, the curves γ1and γ2, and the loop

γaround the focus–focus singularity

since the critical point Pcis of focus–focus type. Note that focus–focus points are positive

by Theorem 3; for a deﬁnition of focus–focus points we refer to [10].

Consider again the curves γ1and γ2shown in Fig. 2. Observe that F−1(γ1)and

F−1(γ2)are invariant under the circle action given by the Hamiltonian ﬂow of J.Let

c1and c2denote the corresponding Chern numbers. By the isotopy, we have that c1=

c(1+ε) and c2=c(1−ε). In particular, c1=c2+1.

Let δ>0 be sufﬁciently small. Consider the following set

S−={x∈F−1(γ1)|J(x)≤jmin +δ},

where jmin is the minimum value of the momentum Jon F−1(γ1). Similarly, we deﬁne

the set

S+={x∈F−1(γ1)|J(x)≥jmax −δ}.

By the construction of the curves γi,thesetsS−and S+are contained in both F−1(γ1)

and F−1(γ2). Topologically, these sets are solid tori.

Let (a−,b−)be two basis cycles on ∂S−such that a−is the meridian and b−is an

orbit of the circle action. Let (a+,b+)be the corresponding cycles on ∂S+. The preimage

F−1(γi)is homeomorphic to the space obtained by gluing these pairs of cycles by

a−

b−=1ci

01

a+

b+,

where ciis the Chern number of F−1(γi). It follows that the monodromy matrix along

γis given by the product

Mγ=1c1

01

1c2

01

−1

.

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Hamiltonian Monodromy and Morse Theory 1381

Since c1=c2+1,we conclude that the monodromy matrix

Mγ=11

01

.

Remark 6 (Fomenko–Zieschang theory). The cycles a±,b±, which we have used when

expressing F−1(γi)as a result of gluing two solid tori, are admissible in the sense of

Fomenko–Zieschang theory [10,32]. It follows, in particular, that the Liouville ﬁbration

of F−1(γi)is determined by the Fomenko–Zieschang invariant (the marked molecule)

A∗ri=∞,ε=1,niA∗

with the n-mark nigiven by the Chern number ci. (The same is true for the regular

energy levels H−1(h).) Therefore, our results show that Hamiltonian monodromy is

also given by the jump in the n-mark. We note that the n-mark and the other labels in the

Fomenko–Zieschang invariant are also deﬁned in the case when no global circle action

exists.

3.2. Geometric monodromy theorem. A common aspect of most of the systems with

non-trivial Hamiltonian monodromy is that the corresponding energy-momentum map

has focus–focus points, which, from the perspective of Morse theory, are saddle points

of the Hamiltonian function.

The following result, which is sometimes referred to as the geometric monodromy

theorem, characterizes monodromy around a focus–focus singularity in systems with

two degrees of freedom.

Theorem 2 (Geometric monodromy theorem, [36,42,43,63]). Monodromy around a

focus–focus singularity is given by the matrix

M=1m

01

,

where m is the number of the focus–focus points on the singular ﬁber.

A related result in the context of the focus–focus singularities is that they come with

a Hamiltonian circle action [63,64].

Theorem 3 (Circle action near focus–focus, [63,64]). In a neighbourhood of a focus–

focus ﬁber,2there exists a unique (up to orientation reversing) Hamiltonian circle action

which is free everywhere except for the singular focus–focus points. Near each singular

point, the momentum of the circle action can be written as

J=1

2(q2

1+p2

1)−1

2(q2

2+p2

2)

for some local canonical coordinates (q1,p1,q2,p2). In particular, the circle action

deﬁnes the anti-Hopf ﬁbration near each singular point.

2That is, a singular ﬁber containing a number of focus–focus points.

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1382 N. Martynchuk, H. W. Broer, K. Efstathiou

Fig. 3. Splitting of the focus–focus singularity; the complexity m=3 in this example

One implication of Theorem 3is that it allows to prove the geometric monodromy

theorem by looking at the circle action. Speciﬁcally, one can apply the Duistermaat–

Heckman theorem in this case; see [64]. A slight modiﬁcation of our argument, used

in the previous Sect. 3.1 to determine monodromy in the spherical pendulum, results in

another proof of the geometric monodromy theorem. We give this proof below.

Proof of Theorem 2By applying integrable surgery, we can assume that the bifurcation

diagram consists of a square of elliptic singularities and a focus–focus singularity in the

middle; see [64]. In the case when there is only one focus–focus point on the singular

focus–focus ﬁber, the proof reduces to the case of the spherical pendulum. Otherwise

the conﬁguration is unstable. Instead of a focus–focus ﬁber with msingular points, one

can consider a new S1-invariant ﬁbration such that it is arbitrary close to the original

one and has msimple (that is, containing only one critical point) focus–focus ﬁbers; see

Fig. 3.

As in the case of the spherical pendulum, we get that the monodromy matrix around

each of the simple focus–focus ﬁbers is given by the matrix

Mi=11

01

.

Since the new ﬁbration is S1-invariant, the monodromy matrix around mfocus–focus

ﬁbers is given by the product of msuch matrices, that is,

Mγ=M1···Mm=1m

01

.

The result follows.

Remark 7 (Duistermaat–Heckman). Consider a symplectic 4-manifold Mand a proper

function Jthat generates a Hamiltonian circle action on this manifold. Assume that

the ﬁxed points are isolated and that the action is free outside these points. From the

Duistermaat–Heckman theorem [22] it follows that the symplectic volume vol(j)of

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Hamiltonian Monodromy and Morse Theory 1383

J−1(j)/S1is a piecewise linear function. Moreover, if j=0 is a critical value with m

positive ﬁxed points of the circle action, then

vol(j)+vol(−j)=2vol(0)−mj.

As was shown in [64], this result implies the geometric monodromy theorem since the

symplectic volume can be viewed as the afﬁne length of the line segment {J=j}in

the image of F. The connection to our approach can be seen from the observation that

the derivative vol(j)coincides with the Chern number of J−1(j). We note that for

the spherical pendulum, the Hamiltonian does not generate a circle action, whereas the

z-component of the angular momentum is not a proper function. Therefore, neither of

these functions can be taken as ‘J’; in order to use the Duistermaat–Heckman theorem,

one needs to consider a local model ﬁrst [64]. Our approach, based on Morse theory, can

be applied directly to the Hamiltonian of the spherical pendulum, even though it does

not generate a circle action.

Remark 8 (Generalization). We observe that even if a simple closed curve γ⊂Rbounds

some complicated arrangement of singularities or, more generally, if the interior of γ

in R2is not contained in the image of the energy-momentum map F, the monodromy

along this curve can still be computed by looking at the energy level Chern numbers.

Speciﬁcally, the monodromy along γis given by

Mγ=1mγ

01

,

where mγ=c(h2)−c(h1)is the difference between the Chern numbers of two (appro-

priately chosen) energy levels.

Remark 9 (Planar scattering). We note that a similar result holds in the case of mechan-

ical Hamiltonian systems on T∗R2that are both scattering and integrable; see [41]. For

such systems, the roles of the compact monodromy and the Chern number are played

by the scattering monodromy and Knauf’s scattering index [34], respectively.

Remark 10 (Many degrees of freedom). The approach presented in this paper depends on

the use of energy-levels and their Chern numbers. For this reason, it cannot be directly

generalized to systems with many degrees of freedom. An approach that admits such a

generalization was developed in [30,40]; we shall recall it in the next section.

3.3. Example: a system with two focus–focus points. Here we illustrate the Morse theory

approach that we developed in this paper on a concrete example of an integrable system

that has more than one focus–focus point. The system was introduced in [55]; it is an

example of a semi-toric system [24,54,60] with a special property that it has two distinct

focus–focus ﬁbers, which are not on the same level of the momentum corresponding to

the circle action.

Let S2be the unit sphere in R3and let ωdenote its volume form, induced from

R3. Take the product S2×S2with the symplectic structure ω⊕2ω. The system in-

troduced in [55] is an integrable system on S2×S2deﬁned in Cartesian coordinates

(x1,y1,z1,x2,y2,z2)∈R3⊕R3by the Poisson commuting functions

H=1

4z1+1

4z2+1

2(x1x2+y1y2)and J=z2+2z2.

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1384 N. Martynchuk, H. W. Broer, K. Efstathiou

Fig. 4. The bifurcation diagram for the system on S2×S2and the loops γ1,γ

2,γ

3around the focus–focus

singularities

The bifurcation diagram of the corresponding energy-momentum map F=(H,J):S2×

S2→R2is shown in Fig. 4.

The system has 4 singular points: two focus–focus and two elliptic–elliptic points.

These singular points are (S,S), (N,S), ( S,N)and (N,N), where Sand Nare the

South and the North poles of S2. Observe that these points are the ﬁxed points of

the circle action generated by the momentum J. The focus–focus points are positive

ﬁxed points (in the sense of Deﬁnition 2) and the elliptic–elliptic points are negative.

Takens’s index theorem implies that the topology of the regular J-levels are S3,S2×S1,

and S3; the corresponding Chern numbers are −1,0,and 1, respectively. Invoking the

argument in Sect. 3.1 for the spherical pendulum (see also Sect. 3.2), we conclude3that

the monodromy matrices along the curves γ1and γ2that encircle the focus–focus points

(see Fig. 4)are

M1=M2=11

01

.(3)

Here the homology basis (a,b)is chosen such that bis an orbit of the circle action.

Remark 11 Observe that the regular H-levels have the following topology: S2×S1,S3,S3,

and S2×S1. We see that the energy levels do not change their topology as the value

of Hpasses the critical value 0, which corresponds to the two focus–focus points. Still,

the monodromy around γ3is nontrivial. Indeed, in view of Eq. (3) and the existence of

a global circle action [19], the monodromy along γ3is given by

M3=M1·M2=12

01

.

3We note that Eq. 3follows also from the geometric monodromy theorem since the circle action gives

a universal sign for the monodromy around the two focus–focus points [19]. Our aim is to prove Eq. 3by

looking at the topology of the energy levels.

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Hamiltonian Monodromy and Morse Theory 1385

The apparent paradox is resolved when one looks at the Chern numbers: the Chern

number of the 3-sphere below the focus–focus points is equal to −1, whereas the Chern

number of the 3-sphere above the focus–focus points is equal to + 1. (The Chern number

of S2×S1is equal to 0 in both cases.) We note that a similar kind of example of an

integrable system for which the monodromy is non-trivial and the energy levels do not

change their topology, is given in [15] (see Burke’s egg (poached)). In the case of Burke’s

egg, the energy levels are non-compact; in the case of the system on S2×S2they are

compact.

4. Symmetry Approach

We note that one can avoid using energy levels by looking directly at the Chern number of

F−1(γ ), where γis the closed curve along which Hamiltonian monodromy is deﬁned.

This point of view was developed in the work [30]. It is based on the following two

results.

Theorem 4 (Fomenko–Zieschang, [10, §4.3.2], [30]). Assume that the energy-momentum

map F is proper and invariant under a Hamiltonian circle action. Let γ⊂image(F)be

a simple closed curve in the set of the regular values of the map F. Then the Hamiltonian

monodromy of the torus bundle F :F−1(γ ) →γis given by

1m

01

∈SL(2,Z),

where m is the Chern number of the principal circle bundle ρ:F−1(γ ) →F−1(γ )/S1,

deﬁned by reducing the circle action.

In the case when the curve γbounds a disk D⊂image(F), the Chern number mcan

be computed from the singularities of the circle action that project into D. Speciﬁcally,

there is the following result.

Theorem 5 ([30]). Let F and γbe as in Theorem 4. Assume that γ=∂D, where

D⊂image(F)is a two-disk, and that the circle action is free everywhere in F −1(D)

outside isolated ﬁxed points. Then the Hamiltonian monodromy of the 2-torus bundle

F:F−1(γ ) →γis given by the number of positive singular points minus the number

of negative singular points in F −1(D).

We note that Theorems 4and 5were generalized to a much more general setting of

fractional monodromy and Seifert ﬁbrations; see [40]. Such a generalization allows one,

in particular, to deﬁne monodromy for circle bundles over 2-dimensional surfaces (or

even orbifolds) of genus g≥1; in the standard case the genus g=1.

Let us now give a new proof of Theorem 4, which makes a connection to the rotation

number. First we shall recall this notion.

We assume that the energy-momentum map Fis invariant under a Hamiltonian circle

action. Without loss of generality, F=(H,J)is such that the circle action is given by the

Hamiltonian ﬂow ϕt

Jof J.LetF−1(f)be a regular torus. Consider a point x∈F−1(f)

and the orbit of the circle action passing through this point. The trajectory ϕt

H(x)leaves

the orbit of the circle action at t=0 and then returns back to the same orbit at some time

T>0. The time Tis called the the ﬁrst return time.Therotation number Θ=Θ( f)

is deﬁned by ϕ2πΘ

J(x)=ϕT

H(x). There is the following result.

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1386 N. Martynchuk, H. W. Broer, K. Efstathiou

Theorem 6 (Monodromy and rotation number, [15]). The Hamiltonian monodromy of

the torus bundle F :F−1(γ ) →γis given by

1m

01

∈SL(2,Z),

where −m is the variation of the rotation number Θ.

Proof First we note that since the ﬂow of Jis periodic on F−1(γ ), the monodromy

matrix is of the form

1m

01

∈SL(2,Z)

for some integer m.

Fix a starting point f0∈γ. Choose a smooth branch of the rotation number Θon

γ\f0and deﬁne the vector ﬁeld XSon F−1(γ \f0)by

XS=T

2πXH−ΘXJ.(4)

By the construction, the ﬂow of XSis periodic. However, unlike the ﬂow of XJ,it is not

globally deﬁned on F−1(γ ). Let α1and α0be the limiting cycles of this vector ﬁeld on

F−1(f0), that is, let α0be given by the ﬂow of the vector ﬁeld XSfor f→f0+ and let

α1be given by the ﬂow of XSfor f→f0−. Then

α1=α0+mbf0,

where −mis the variation of the rotation number along γ. Indeed, if the variation of the

rotation number is −m, then the vector ﬁeld T(f0)

2πXH−Θ( f0)XJon F−1(f0)changes

to T(f0)

2πXH−(Θ( f0)−m)XJafter ftraverses γ. Since α1is the result of the parallel

transport of α0along γ, we conclude that m=m. The result follows.

We are now ready to prove Theorem 4.

Proof Take an invariant metric gon F−1(γ ) and deﬁne a connection 1-form σof the

principal S1bundle ρ:Eγ→Eγ/S1as follows:

σ(XJ)=iand σ(XH)=σ(e)=0,

where eis orthogonal to XJand XHwith respect to the metric g. Since the ﬂows ϕt

H

and ϕτ

Jcommute, σis indeed a connection one-form.

By the construction,

i

2πα0

σ−α1

σ=−

im

2πbf0

σ=m.

Since α0α1bounds a cylinder C⊂F−1(γ \f0), we also have

m=i

2πC

dσ=Eγ/S1c1,

where c1is the Chern class of the circle bundle ρ:Eγ→Eγ/S1. The result follows.

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Hamiltonian Monodromy and Morse Theory 1387

5. Discussion

In this paper we studied Hamiltonian monodromy in integrable two-degree of freedom

Hamiltonian systems with a circle action. We showed how Takens’s index theorem,

which is based on Morse theory, can be used to compute Hamiltonian monodromy. In

particular, we gave a new proof of the monodromy around a focus–focus singularity using

the Morse theory approach. An important implication of our results is a connection of the

geometric theory developed in the works [29,40] to Cushman’s argument, which is also

based on Morse theory. New connections to the rotation number and to Duistermaat–

Heckman theory were also discussed.

Acknowledgement. We would like to thank Prof. A. Bolsinov and Prof. H. Waalkensfor useful and stimulating

discussions. We would also like to thank the anonymous referee for his suggestions for improvement.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Inter-

national License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,

and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source,

provide a link to the Creative Commons license, and indicate if changes were made.

Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims

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A Hamiltonian monodromy

A typical situation in which monodromy arises is the case of an integrable system on a

4-dimensional symplectic manifold (M4,Ω). Such a system is speciﬁed by the energy-

momentum (or the integral)map

F=(H,J):M→R2.

Here His the Hamiltonian of the system and the momentum Jis a ‘symmetry’ function,

that is, the Poisson bracket

{H,J}=Ω−1(dJ,dH)=0

vanishes. We will assume that the map Fis proper, that is, that preimages of compact

sets are compact, and that the ﬁbers F−1(f)of Fare connected. Then near any regular

value of Fthe functions Hand Jcan be combined into new functions I1=I1(H,J)

and I2=I2(H,J)such that the symplectic form has the canonical form

Ω=dI

1∧dϕ1+dI

2∧dϕ2

for some angle coordinates ϕ1,ϕ

2on the ﬁbers of F. This follows from the Arnol’d–

Liouville theorem [3]. We note that the regular ﬁbers of Fare tori and that the motion

on these tori is quasi-periodic.

The coordinates Iithat appear in the Arnol’d–Liouville theorem are called action

coordinates. It can be shown that if pdq is a local primitive 1-from of the symplectic

form, then these coordinates are given by the formula

Ii=

αi

pdq,(5)

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1388 N. Martynchuk, H. W. Broer, K. Efstathiou

where αi,i=1,2,are two independent cycles on an Arnol’d–Liouville torus. However,

this formula is local even if the symplectic form Ωis exact. The reason for this is that

the cycles αican not, generally speaking, be chosen for each torus F−1(f)in a such a

way that the maps f→ αi(f)are continuous at all regular values fof F.Thisisthe

essence of Hamiltonian monodromy. Speciﬁcally, it is deﬁned as follows.

Let R⊂image(F)be the set of the regular values of F. Consider the restriction map

F:F−1(R)→R.

We observe that this map is a torus bundle: locally it is a direct product Dn×Tn,the

trivialization being achieved by the action-angle coordinates. Hamiltonian monodromy

is deﬁned as a representation

π1(R,f0)→Aut H1(F−1(f0))

of the fundamental group π1(R,f0)in the group of automorphisms of the integer ho-

mology group H1(F−1(f0)). Each element γ∈π1(R,f0)acts via parallel transport of

integer homology cycles αi;see[20].

We note that the appearance of the homology groups is due to the fact that the action

coordinates (5) depend only on the homology class of αion the Arnol’d–Liouville torus.

We observe that since the ﬁbers of Fare tori, the group H1(F−1(f0)) is isomorphic to

Z2. It follows that the monodromy along a given path γis characterized by an integer

matrix Mγ∈GL(2,Z), called the monodromy matrix along γ. It can be shown that the

determinant of this matrix equals 1.

Remark 12 (Examples and generalizations). Non-trivial monodromy has been observed

in various examples of integrable systems, including the most fundamental ones, such

as the spherical pendulum [15,20], the hydrogen atom in crossed ﬁelds [18] and the

spatial Kepler problem [26,39]. This invariant has also been generalized in several dif-

ferent directions, leading to the notions of quantum [16,59], fractional [28,40,48] and

scattering [5,25,29,39]monodromy.

Remark 13 (Topological deﬁnition of monodromy). Topologically, one can deﬁne Hamil-

tonian monodromy along a loop γas monodromy of the torus (in the non-compact case

— cylinder) bundle over this loop. More precisely, consider a T2-torus bundle

F:F−1(γ ) →γ, γ =S1.

It can be obtained from a trivial bundle [0,2π]×T2by gluing the boundary tori via a

homeomorphism f, called the monodromy of F. In the context of integrable systems

(when Fis the energy-momentum map and γis a loop in the set of the regular values)

the matrix of the push-forward map

f:H1(T2)→H1(T2)

coincides with the monodromy matrix along γin the above sense. It follows, in particular,

that monodromy can be deﬁned for any torus bundle.

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Hamiltonian Monodromy and Morse Theory 1389

B Chern classes

Let Mbe an S1-invariant submanifold of Mwhich does not contain the critical points

of H. The circle action on Mis then free and we have a principal circle bundle

ρ:M→M/S1.

Let XJdenote the vector ﬁeld on Mcorresponding to the circle action (such that the

ﬂow of XJgives the circle action) and let σbe a 1-form on Msuch that the following

two conditions hold

(i) σ(XJ)=iand (ii) R∗

g(σ ) =σ.

Here i∈iR— the Lie algebra of S1={eiϕ∈C|ϕ∈[0,2π]} and Rgis the (right)

action of S1.

The Chern (or the Euler) class4can then deﬁned as

c1=s(idw/2π) ∈H2(M/S1,R),

where sis any local section of the circle bundle ρ:M→M/S1.Here H2(M/S1,R)

stands for the second de Rham cohomology group of the quotient M/S1.

We note that if the manifold Mis compact and 3-dimensional, the Chern number of

M(see Deﬁnition 1) is equal to the integral

M/S1c1

of the Chern class c1over the base manifold M/S1.

A non-trivial example of a circle bundle with non-trivial Chern class is given by the

(anti-)Hopf ﬁbration. Recall that the Hopf ﬁbration of the 3-sphere

S3={(z,w)∈C2|1=|z|2+|w|2}

is the principal circle bundle S3→S2obtained by reducing the circle action (z,w) →

(eitz,eit w). The circle action (z,w) → (e−itz,eitw) deﬁnes the anti-Hopf ﬁbration of

S3.

Lemma 2 The Chern number of the Hopf ﬁbration is equal to −1, while for the anti-Hopf

ﬁbration it is equal to 1.

Proof Consider the case of the Hopf ﬁbration (the anti-Hopf case is analogous). Its

projection map h:S3→S2is deﬁned by h(z,w) =(z:w) ∈CP1=S2.Put

U1={(u:1)|u∈C,|u|<1}and U2={(1:v) |v∈C,|v|<1}.

Deﬁne the section sj:Uj→S3by the formulas

s1((u:1)) =u

|u|2+1,1

|u|2+1

4This Chern class should not be confused with Duistermaat’s Chern class, which is another obstruction to

the existence of global action-angle coordinates; see [20,38].

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1390 N. Martynchuk, H. W. Broer, K. Efstathiou

and

s2((1:v)) =1

|v|2+1,v

|v|2+1.

Now, the gluing cocycle t12 :S1=U1∩U2→S1corresponding to the sections s1and

s2is given by

t12((u:1)) =exp (−iArg u).

If follows that the winding number equals −1 (the loop αin Deﬁnition 1is given by the

equator S1=U1∩U2in this case).

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