ArticlePDF Available

Hamiltonian Monodromy and Morse Theory

Authors:

Abstract and Figures

We show that Hamiltonian monodromy of an integrable two degrees of freedom 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 specifies how the energy-h Chern number changes when h passes 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.
This content is subject to copyright. Terms and conditions apply.
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
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 H1(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
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
xixj
(x)
is non-degenerate. We shall assume that His a proper1function and that it is invariant
under a smooth circle action G:M×S1Mthat 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={xM|H(x)=h},the circle action gives rise to the
circle bundle
ρh:HhBh=Hh/S1.
By deﬁnition, the ﬁbers ρ1
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:HhBhhas an
‘almost global’ section
s:Bhρ1
h(Bh)
that is not deﬁned at most in one point bbh.Let αbe a (small) loop that encircles this
point.
1Preimages of compact sets are compact.
1376 N. Martynchuk, H. W. Broer, K. Efstathiou
Deﬁnition 1. The Chern number c(h)of the principal bundle
ρh:HhBh
can be deﬁned as the winding number of s(α) along the orbit ρ1
h(b). In other words,
c(h)is the degree of the map
S1=αs(α) ρ1
h(b)=S1,
where the map s(α) ρ1
h(b)is induced by a retraction of a tubular neighbourhood of
ρ1
h(b)onto ρ1
h(b).
Remark 2. We note that the Chern number c(h)is a topological invariant of the bundle
ρh:HhBhwhich 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), tS1,
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), tS1,
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) → (eitz,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) → (eitz,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(hcε) ±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.
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 H1(−∞,hc+ε]can be obtained
from the manifold H1(−∞,hcε]by attaching a handle Dλ×D4λ, where λis
the index of the critical point on the level H1(hc). More speciﬁcally, for a suitable
neighbourhood Dλ×D4λMof the critical point (with Dmstanding for an m-
dimensional ball), H1(−∞,hc+ε]deformation retracts onto the set
X=H1(−∞,hcε]∪Dλ×D4λ
and, moreover,
H1(−∞,hc+ε]X=H1(−∞,hcε]∪Dλ×D4λ(1)
up to a diffeomorphism. We note that by the construction, the intersection of the handle
Dλ×D4λwith H1(−∞,hcε]is the subset Sλ1×D4λH1(hcε);see
[44]. For simplicity, we shall assume that the handle is disjoint from H1(hc+ε).By
taking the boundary in Eq. (1), we get that
H1(hc+ε) X=(H1(hcε)\Sλ1×D4λ)Dλ×S4λ1.(2)
Here the union (Dλ×S4λ1)(Sλ1×D4λ)is the boundary S3=∂(Dλ×D4λ)
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 H1(hc+ε) and H1(hcε) 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(hcε) +c(S3)=c(hcε) ±1.
It is left to show that c(hc+ε) =c(∂ X)(we note that even though we know that
H1(hc+ε) and Xare diffeomorphic, we cannot yet conclude that they have the same
Chern numbers).
Let the subset YMbe deﬁned as the closure of the set
H1[hcε, hc+ε]\Dλ×D4λ.
We observe that Yis a compact submanifold of Mand that Y=XH1(hc+ε),
that is, Yis a cobordism in Mbetween Xand H1(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:YY/S1, where ρis the reduction map; see Appendix B. This concludes
the proof of the theorem.
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(hcε) +
k
i=1
sk,
where sk1 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
(TS2|TS2,H|TS2), 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=xpyypx(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):TS2R2,
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 Rimage(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 f0R. 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 TS2, any orbit of this action on F1(γ (0))
can be transported along γ.Let(a,b)be a basis of H1(F1(γ (0))), where bis given by
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
TS2. From the Morse lemma, we have that H1(1ε), ε (0,2), is diffeomorphic
to the 3-sphere S3. On the other hand, H1(1+ε) is diffeomorphic to the unit cotangent
bundle T
1S2. It follows that the monodromy index mγ= 0. Indeed, the energy levels
H1(1+ε) and H1(1ε) are isotopic, respectively, to F11)and F12), where
γ1and γ2are the curves shown in Fig. 2.Ifmγ=0, then the preimages F11)and
F12)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(1ε) +1
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 F11)and
F12)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(1ε). In particular, c1=c2+1.
Let δ>0 be sufﬁciently small. Consider the following set
S={xF11)|J(x)jmin +δ},
where jmin is the minimum value of the momentum Jon F11). Similarly, we deﬁne
the set
S+={xF11)|J(x)jmax δ}.
By the construction of the curves γi,thesetsSand S+are contained in both F11)
and F12). Topologically, these sets are solid tori.
Let (a,b)be two basis cycles on Ssuch that ais the meridian and bis an
orbit of the circle action. Let (a+,b+)be the corresponding cycles on S+. The preimage
F1i)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 F1i). It follows that the monodromy matrix along
γis given by the product
Mγ=1c1
01
1c2
01
1
.
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 F1i)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 F1i)is determined by the Fomenko–Zieschang invariant (the marked molecule)
Ari=∞=1,niA
with the n-mark nigiven by the Chern number ci. (The same is true for the regular
energy levels H1(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.
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
Hamiltonian Monodromy and Morse Theory 1383
J1(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 J1(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 γRbounds
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 TR2that 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 ω2ω. The system in-
troduced in [55] is an integrable system on S2×S2deﬁned in Cartesian coordinates
(x1,y1,z1,x2,y2,z2)R3R3by the Poisson commuting functions
H=1
4z1+1
4z2+1
2(x1x2+y1y2)and J=z2+2z2.
1384 N. Martynchuk, H. W. Broer, K. Efstathiou
Fig. 4. The bifurcation diagram for the system on S2×S2and the loops γ1
2
3around the focus–focus
singularities
The bifurcation diagram of the corresponding energy-momentum map F=(H,J):S2×
S2R2is 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.
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
F1(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 γimage(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 :F1(γ ) γis given by
1m
01
SL(2,Z),
where m is the Chern number of the principal circle bundle ρ:F1(γ ) F1(γ )/S1,
deﬁned by reducing the circle action.
In the case when the curve γbounds a disk Dimage(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
Dimage(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:F1(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 g1; 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.LetF1(f)be a regular torus. Consider a point xF1(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.
1386 N. Martynchuk, H. W. Broer, K. Efstathiou
Theorem 6 (Monodromy and rotation number, [15]). The Hamiltonian monodromy of
the torus bundle F :F1(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 F1(γ ), the monodromy
matrix is of the form
1m
01
SL(2,Z)
for some integer m.
Fix a starting point f0γ. Choose a smooth branch of the rotation number Θon
γ\f0and deﬁne the vector ﬁeld XSon F1\f0)by
XS=T
2πXHΘXJ.(4)
By the construction, the ﬂow of XSis periodic. However, unlike the ﬂow of XJ,it is not
globally deﬁned on F1(γ ). Let α1and α0be the limiting cycles of this vector ﬁeld on
F1(f0), that is, let α0be given by the ﬂow of the vector ﬁeld XSfor ff0+ and let
α1be given by the ﬂow of XSfor ff0. 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)
2πXHΘ( f0)XJon F1(f0)changes
to T(f0)
2πXH( 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 F1(γ ) and deﬁne a connection 1-form σof the
principal S1bundle ρ:EγEγ/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
σα1
σ=−
im
2πbf0
σ=m.
Since α0α1bounds a cylinder CF1\f0), we also have
m=i
2πC
dσ=Eγ/S1c1,
where c1is the Chern class of the circle bundle ρ:EγEγ/S1. The result follows.
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.
and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source,
Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims
in published maps and institutional afﬁliations.
A Hamiltonian monodromy
A typical situation in which monodromy arises is the case of an integrable system on a
4-dimensional symplectic manifold (M4). Such a system is speciﬁed by the energy-
momentum (or the integral)map
F=(H,J):MR2.
Here His the Hamiltonian of the system and the momentum Jis a ‘symmetry’ function,
that is, the Poisson bracket
{H,J}=Ω1(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 F1(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
1dϕ1+dI
2dϕ2
for some angle coordinates ϕ1
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)
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 F1(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 Rimage(F)be the set of the regular values of F. Consider the restriction map
F:F1(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(F1(f0))
of the fundamental group π1(R,f0)in the group of automorphisms of the integer ho-
mology group H1(F1(f0)). Each element γπ1(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(F1(f0)) is isomorphic to
Z2. It follows that the monodromy along a given path γis characterized by an integer
matrix MγGL(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:F1(γ ) γ, γ =S1.
It can be obtained from a trivial bundle [0,2πT2by 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.
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:MM/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 iiR— the Lie algebra of S1={eiϕC|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:MM/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 S3S2obtained by reducing the circle action (z,w) →
(eitz,eit w). The circle action (z,w) → (eitz,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:S3S2is deﬁned by h(z,w) =(z:w) CP1=S2.Put
U1={(u:1)|uC,|u|<1}and U2={(1:v) |vC,|v|<1}.
Deﬁne the section sj:UjS3by 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].
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=U1U2S1corresponding 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=U1U2in this case).
References
1. Arnol’d, V.I.: Proof of a theorem of A.N. Kolmogorov on the invariance of quasi-periodic motions under
small perturbations of the Hamiltonian. Russ. Math. Surv. 18(5), 9–36 (1963)
2. Arnol’d, V.I.: Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, vol. 60.
Springer, New York (translated by K. Vogtmann and A, Weinstein, 1978)
3. Arnol’d, V.I., Avez, A.: Ergodic Problems of Classical Mechanics. W.A. Benjamin Inc, New York (1968)
4. Audin, M.: Torus Actions on Symplectic Manifolds. Birkhäuser, Basel (2004)
5. Bates, L., Cushman, R.: Scattering monodromy and the A1 singularity. Cent. Eur. J. Math. 5(3), 429–451
(2007)
6. Bates, L.M.: Monodromy in the champagne bottle. J. Appl. Math. Phys. 42(6), 837–847 (1991)
7. Bates, L.M., Zou, M.: Degeneration of Hamiltonian monodromy cycles. Nonlinearity 6(2), 313–335
(1993)
8. Binney, J., Tremaine, S.: Galactic Dynamics. Princeton University Press, Princetion (1987)
9. Bochner, S.: Compact groups of differentiable transformations. Ann. Math. 46(3), 372–381 (1945)
10. Bolsinov, A.V., Fomenko, A.T.: Integrable Hamiltonian Systems: Geometry, Topology, Classiﬁcation.
CRC Press, Boca Raton (2004)
11. Broer, H.W., Cushman, R.H., Fassò, F., Takens, F.: Geometry of KAM tori for nearly integrable Hamil-
tonian systems. Ergod. Theory Dyn. Syst. 27(3), 725–741 (2007)
12. Broer, H.W., Huitema, G.B., Sevryuk, M.B.: Quasi-Periodic Motions in Families of Dynamical Systems:
Order Amidst Chaos. Lecture Notes in Mathematics, vol. 1645. Springer, Berlin (1996)
13. Broer, H.W., Takens, F.: Unicity of KAM tori. Ergod. Theory Dyn. Syst. 27(3), 713–724 (2007)
14. Child, M.S.: Quantum states in a champagne bottle. J. Phys. A: Math. Gen. 31(2), 657–670 (1998)
15. Cushman, R.H., Bates, L.M.: Global Aspects of Classical Integrable Systems, 2nd edn. Birkhäuser, Basel
(2015)
16. Cushman, R.H., Duistermaat, J.J.: The quantum mechanical spherical pendulum. Bull. Am. Math. Soc.
19(2), 475–479 (1988)
17. Cushman, R.H., Knörrer, H.: The Energy Momentum Mapping of the Lagrange Top, Differential Geo-
metric Methods in Mathematical Physics. Lecture Notes in Mathematics, vol. 1139, pp. 12–24. Springer,
Berlin (1985)
18. Cushman, R.H., Sadovskií, D.A.: Monodromy in the hydrogen atom in crossed ﬁelds. Physica D 142(1–2),
166–196 (2000)
19. Cushman, R.H., V˜uNgo
.c, S.: Sign of the monodromy for Liouville integrable systems. Ann. Henri
Poincaré 3(5), 883–894 (2002)
20. Duistermaat, J.J.: On global action-angle coordinates. Commun. Pure Appl. Math. 33(6), 687–706 (1980)
21. Duistermaat, J.J.: The monodromy in the Hamiltonian Hopf bifurcation. Z. Angew. Math. Phys. 49(1),
156 (1998)
22. Duistermaat, J.J., Heckman, G.J.: On the variation in the cohomology of the symplectic form of the
reduced phase space. Invent. Math. 69(2), 259–268 (1982)
23. Dullin, H.R., Pelayo, Á.: Generating hyperbolic singularities in semitoric systems via Hopf bifurcations.
J. Nonlinear Sci. 26, 787–811 (2016)
24. Dullin, H.R., Pelayo, Á.: Generating hyperbolic singularities in semitoric systems via Hopf bifurcations.
J. Nonlinear Sci. 26(3), 787–811 (2016)
25. Dullin, H.R., Waalkens, H.: Nonuniqueness of the phase shift in central scattering due to monodromy.
Phys. Rev. Lett. 101, 070405 (2008)
Hamiltonian Monodromy and Morse Theory 1391
26. Dullin, H.R., Waalkens, H.: Defect in the joint spectrum of hydrogen due to monodromy. Phys. Rev. Lett.
120, 020507 (2018)
27. Efstathiou, K.: Metamorphoses of Hamiltonian Systems with Symmetries. Springer, Berlin (2005)
28. Efstathiou, K., Broer, H.W.: Uncovering fractional monodromy. Commun. Math. Phys. 324(2), 549–588
(2013)
29. Efstathiou, K., Giacobbe, A., Mardeši´c, P., Sugny, D.: Rotation forms and local Hamiltonian monodromy,
Submitted (2016)
30. Efstathiou, K., Martynchuk, N.: Monodromy of Hamiltonian systems with complexity-1 torus actions.
Geom. Phys. 115, 104–115 (2017)
31. Fomenko, A.T., Matveev, S.V.: Algorithmic and Computer Methods for Three-Manifolds, 1st edn.
Springer, Dordrecht (1997)
32. Fomenko, A.T., Zieschang, H.: Topological invariant and a criterion for equivalence of integrable Hamil-
tonian systems with two degrees of freedom. Izv. Akad. Nauk SSSR Ser. Mat. 54(3), 546–575 (1990).
(in Russian)
33. Jaynes, E.T., Cummings, F.W.: Comparison of quantum and semiclassical radiation theories with appli-
cation to the beam maser. Proc. IEEE 51(1), 89–109 (1963)
34. Knauf, A.: Qualitative aspects of classical potential scattering. Regul. Chaotic Dyn. 4(1), 3–22 (1999)
35. Kolmogorov, A.N.: Preservation of conditionally periodic movements with small change in the Hamilton
function. Dokl. Akad. Nauk SSSR 98, 527 (1954)
36. Lerman, L.M., Umanski˘ı, Y.L.: Classiﬁcation of four-dimensional integrable Hamiltonian systems and
Poisson actions of R2in extended neighborhoods of simple singular points I. Sb. Math. 77(2), 511–542
(1994)
37. Liouville, J.: Note sur l’intégration des équations différentielles de la dynamique, présentée au Bureau
des Longitudes le 29 juin 1853. Journal de mathématiques pures et appliquées 20, 137–138 (1855)
38. Lukina, O.V., Takens, F., Broer, H.W.: Global properties of integrable Hamiltonian systems. Regul.
Chaotic Dyn. 13(6), 602–644 (2008)
39. Martynchuk, N., Dullin, H.R., Efstathiou, K., Waalkens, H.: Scattering invariants in Euler’s two-center
problem. Nonlinearity 32(4), 1296–1326 (2019)
40. Martynchuk, N., Efstathiou, K.: Parallel transport along Seifert manifolds and fractional monodromy.
Commun. Math. Phys. 356(2), 427–449 (2017)
41. Martynchuk, N., Waalkens, H.: Knauf’s degree and monodromy in planar potential scattering. Regul.
Chaotic Dyn. 21(6), 697–706 (2016)
42. Matsumoto, Y.: Topology of torus ﬁbrations. Sugaku Expo. 2, 55–73 (1989)
43. Matveev, V.S.: Integrable Hamiltonian system with two degrees of freedom. The topological structure of
saturated neighbourhoods of points of focus–focus and saddle–saddle type. Sb. Math. 187(4), 495–524
(1996)
44. Milnor, J.: Morse theory. Princeton University Press, Princeton (1963)
45. Milnor, J.W., Stasheff, J.D.: Characteristic Classes. Princeton University Press, Princeton (1974)
46. Mineur, H.: Réduction des systèmes mécaniques à ndegré de liberté admettant nintégrales premières
uniformes en involution aux systèmes à variables séparées. J. Math. Pure Appl. IX Sér. 15, 385–389
(1936)
47. Moser, J.: Convergent series expansions for quasi-periodic motions. Math. Ann.169(1), 136–176 (1967)
48. Nekhoroshev, N.N., Sadovskií, D.A., Zhilinskií, B.I.: Fractional Hamiltonian monodromy. Ann. Henri
Poincaré 7, 1099–1211 (2006)
49. Pelayo, A., Vu Ngoc, S.: Hamiltonian dynamical and spectral theory for spin-oscillators. Commun. Math.
Phys. 309(1), 123–154 (2012)
50. Postnikov, M.M.: Differential Geometry IV. MIR, Moscow (1982)
51. Rink, B.W.: A Cantor set of tori with monodromy near a focus–focus singularity. Nonlinearity 17(1),
347–356 (2004)
52. Sadovskií, D.A., Zhilinskií, B.I.: Monodromy, diabolic points, and angular momentum coupling. Phys.
Lett. A 256(4), 235–244 (1999)
53. Salamon, D.A.: The Kolmogorov–Arnold–Moser theorem. Math. Phys. Electron. J. 10(3), 1–37 (2004)
54. Sepe, D., Sabatini, S., Hohloch, S.: From compact semi-toric systems to Hamiltonian S-1-spaces, vol.
35, pp. 247–281 (2014)
55. Sonja, H., Palmer, J.: A family of compact semitoric systems with two focus–focus singularities. J. Geom.
Mech. 10(3), 331–357 (2018)
56. Strominger, A., Yau, S.-T., Zaslow, E.: Mirror symmetry is T-duality. Nucl. Phys. B 479(1), 243–259
(1996)
57. Takens, F.: Private communication (2010)
58. Urbantke, H.K.: The Hopf ﬁbration—seven times in physics. J. Geom. Phys. 46(2), 125–150 (2003)
59. V˜uNgo
.c, S.: Quantum monodromy in integrable systems. Commun. Math. Phys. 203(2), 465–479 (1999)
1392 N. Martynchuk, H. W. Broer, K. Efstathiou
60. V˜uNgo
.c, S.: Moment polytopes for symplectic manifolds with monodromy. Adv. Math. 208(2), 909–934
(2007)
61. Waalkens, H., Dullin, H.R., Richter, P.H.: The problem of two ﬁxed centers: bifurcations, actions, mon-
odromy. Physica D 196(3–4), 265–310 (2004)
62. Wasserman, A.G.: Equivariant differential topology. Topology 8(2), 127–150 (1969)
63. Zung, N.T.: A note on focus–focus singularities. Differ. Geom. Appl. 7(2), 123–130 (1997)
64. Zung, N.T.: Another note on focus–focus singularities. Lett. Math. Phys. 60(1), 87–99 (2002)
Communicated by J. Marklof
1.
2.
3.
4.
5.
6.
Terms and Conditions
Springer Nature journal content, brought to you courtesy of Springer Nature Customer Service Center
GmbH (“Springer Nature”).
Springer Nature supports a reasonable amount of sharing of research papers by authors, subscribers
and authorised users (“Users”), for small-scale personal, non-commercial use provided that all
copyright, trade and service marks and other proprietary notices are maintained. By accessing,
sharing, receiving or otherwise using the Springer Nature journal content you agree to these terms of
use (“Terms”). For these purposes, Springer Nature considers academic use (by researchers and
students) to be non-commercial.
These Terms are supplementary and will apply in addition to any applicable website terms and
conditions, a relevant site licence or a personal subscription. These Terms will prevail over any
conflict or ambiguity with regards to the relevant terms, a site licence or a personal subscription (to
the extent of the conflict or ambiguity only). For Creative Commons-licensed articles, the terms of
the Creative Commons license used will apply.
We collect and use personal data to provide access to the Springer Nature journal content. We may
also use these personal data internally within ResearchGate and Springer Nature and as agreed share
it, in an anonymised way, for purposes of tracking, analysis and reporting. We will not otherwise
disclose your personal data outside the ResearchGate or the Springer Nature group of companies
While Users may use the Springer Nature journal content for small scale, personal non-commercial
use, it is important to note that Users may not:
use such content for the purpose of providing other users with access on a regular or large scale
basis or as a means to circumvent access control;
use such content where to do so would be considered a criminal or statutory offence in any
jurisdiction, or gives rise to civil liability, or is otherwise unlawful;
falsely or misleadingly imply or suggest endorsement, approval , sponsorship, or association
unless explicitly agreed to by Springer Nature in writing;
use bots or other automated methods to access the content or redirect messages
override any security feature or exclusionary protocol; or
share the content in order to create substitute for Springer Nature products or services or a
systematic database of Springer Nature journal content.
In line with the restriction against commercial use, Springer Nature does not permit the creation of a
product or service that creates revenue, royalties, rent or income from our content or its inclusion as
part of a paid for service or for other commercial gain. Springer Nature journal content cannot be
used for inter-library loans and librarians may not upload Springer Nature journal content on a large
scale into their, or any other, institutional repository.
These terms of use are reviewed regularly and may be amended at any time. Springer Nature is not
obligated to publish any information or content on this website and may remove it or features or
functionality at our sole discretion, at any time with or without notice. Springer Nature may revoke
this licence to you at any time and remove access to any copies of the Springer Nature journal content
which have been saved.
To the fullest extent permitted by law, Springer Nature makes no warranties, representations or
guarantees to Users, either express or implied with respect to the Springer nature journal content and
all parties disclaim and waive any implied warranties or warranties imposed by law, including
merchantability or fitness for any particular purpose.
Please note that these rights do not automatically extend to content, data or other material published
by Springer Nature that may be licensed from third parties.
If you would like to use or distribute our Springer Nature journal content to a wider audience or on a
Nature at
onlineservice@springernature.com
... We then explain a dynamical manifestation of non-trivial Hamiltonian monodromy. Afterwards, we come back to the spherical pendulum and discuss the monodromy from a different point of view based on Morse theory and Chern numbers (a general situation is treated in the work [56]). We conclude this section with an extension of Hamiltonian monodromy to nearly integrable systems. ...
... For example, one can apply the Duistermaat-Heckman theorem; see [86]. A related and purely topological proof will be given below on the example of the spherical pendulum, following the point of view of [39,55,56,58]. For other approaches to the geometric monodromy theorem, we refer the reader to [5,23,38,79]. ...
... The spherical pendulum. We now come back to the case of the spherical pendulum and prove that the monodromy matrix of this system is given by Eq. 2. We shall mainly focus on a topological idea which goes back to R. Cushman and F. Takens and which has been developed in the works [39,56,58]. ...
Preprint
The notion of monodromy was introduced by J. J. Duistermaat as the first obstruction to the existence of global action coordinates in integrable Hamiltonian systems. This invariant was extensively studied since then and was shown to be non-trivial in various concrete examples of finite-dimensional integrable systems. The goal of the present paper is to give a brief overview of monodromy and discuss some of its generalisations. In particular, we will discuss the monodromy around a focus-focus singularity and the notions of quantum, fractional and scattering monodromy. The exposition will be complemented with a number of examples and open problems.
... We then explain a dynamical manifestation of non-trivial Hamiltonian monodromy. Afterwards, we come back to the spherical pendulum and discuss the monodromy from a different point of view based on Morse theory and Chern numbers (a general situation is treated in the work [58]). We conclude this section with an extension of Hamiltonian monodromy to nearly integrable systems. ...
... For example, one can apply the Duistermaat-Heckman theorem; see [91]. A related and purely topological proof will be given below on the example of the spherical pendulum, following the point of view of [39,57,58,60]. For other approaches to the geometric monodromy theorem, we refer the reader to [5,23,38,83]. ...
... The spherical pendulum. We now come back to the case of the spherical pendulum and prove that the monodromy matrix of this system is given by Eq. 2. We shall mainly focus on a topological idea which goes back to R. Cushman and F. Takens and which has been developed in the works [39,58,60]. ...
Article
The notion of monodromy was introduced by J.J. Duistermaat as the first obstruction to the existence of global action coordinates in integrable Hamiltonian systems. This invariant was extensively studied since then and was shown to be non-trivial in various concrete examples of finite-dimensional integrable systems. The goal of the present paper is to give a brief overview of monodromy and discuss some of its generalisations. In particular, we will discuss the monodromy around a focus-focus singularity and the notions of quantum, fractional and scattering monodromy. The exposition will be complemented with a number of examples and open problems.
... The computation of this function and its variation are therefore the building blocks for describing HM [7,8,33]. Note that more geometric approaches based on the global structure of the torus bundle can also be used to show the non-trivial Monodromy of a Hamiltonian system [34]. ...
Preprint
Full-text available
Hamiltonian Monodromy is the simplest topological obstruction to the existence of global action-angle coordinates in a completely integrable system. We show that this property can be studied in a neighborhood of a focus-focus singularity by a spectral Lax pair approach. From the Lax pair, we derive a Riemann surface which allows us to compute in a straightforward way the corresponding Monodromy matrix. The general results are applied to the Jaynes-Cummings model and the spherical pendulum.
Preprint
Classical Morse theory proceeds by considering sublevel sets $f^{-1}(-\infty, a]$ of a Morse function $f: M \to R$, where $M$ is a smooth finite-dimensional manifold. In this paper, we study the topology of the level sets $f^{-1}(a)$ and give conditions under which the topology of $f^{-1}(a)$ changes when passing a critical value. We show that for a general class of functions, which includes all exhaustive Morse function, the topology of a regular level $f^{-1}(a)$ always changes when passing a single critical point, unless the index of the critical point is half the dimension of the manifold $M$. When $f$ is a natural Hamiltonian on a cotangent bundle, we obtain more precise results in terms of the topology of the configuration space. (Counter-)examples and applications to celestial mechanics are also discussed.
Article
Full-text available
The problem of two fixed centers was introduced by Euler as early as in 1760. It plays an important role both in celestial mechanics and in the microscopic world. In the present paper we study the spatial problem in the case of arbitrary (both positive and negative) strengths of the centers. Combining techniques from scattering theory and Liouville integrability, we show that this spatial problem has topologically non-trivial scattering dynamics, which we identify as scattering monodromy. The approach that we introduce in this paper applies more generally to scattering systems that are integrable in the Liouville sense.
Article
Full-text available
In addition to the well known case of spherical coordinates the hydrogen atom separates in three further coordinate systems. Separating in a particular coordinate system defines a system of three commuting operators. We show that the joint spectrum of the Hamilton operator, and the $z$-components of the angular momentum and quantum Laplace-Runge-Lenz vectors obtained from separation in prolate spheroidal coordinates has quantum monodromy for energies sufficiently close to the ionization threshold. This means that one cannot globally assign quantum numbers to the joint spectrum. Whereas the principal quantum number $n$ and the magnetic quantum number $m$ correspond to the Bohr-Sommerfeld quantization of globally defined classical actions a third quantum number cannot be globally defined because the third action is globally multi valued.
Article
Full-text available
The notion of fractional monodromy was introduced by Nekhoroshev, Sadovskií and Zhilinskií as a generalization of standard (‘integer’) monodromy in the sense of Duistermaat from torus bundles to singular torus fibrations. In the present paper we prove a general result that allows one to compute fractional monodromy in various integrable Hamiltonian systems. In particular, we show that the non-triviality of fractional monodromy in 2 degrees of freedom systems with a Hamiltonian circle action is related only to the fixed points of the circle action. Our approach is based on the study of a specific notion of parallel transport along Seifert manifolds.
Book
Integrable Hamiltonian systems have been of growing interest over the past 30 years and represent one of the most intriguing and mysterious classes of dynamical systems. This book explores the topology of integrable systems and the general theory underlying their qualitative properties singularites and topological invariants. The authors both of whom have contributed significantly to the field develop the classification theory for integrable systems with two degrees of freedom. This theory allows one to distinguish such systems up to two natural equivalence relations: the equivalence of the associated foliation into Liouville tori and the usual orbital equaivalence. The authors show that in both cases one can find complete sets of invariants that give the solution of the classification problem. The first part of the book systematically presents the general construction of these invariants including many examples and applications. In the second part the authors apply the general methods of the classification theory to the classical integrable problems in rigid body dynamics and describe their topological portraits bifurcations of Liouville tori and local and global topological invariants. They show how the classification theory helps find hidden isomorphisms between integrable systems and present as an example their proof that two famous systems--the Euler case in rigid body dynamics and the Jacobi problem of geodesics on the ellipsoid--are orbitally equivalent. Integrable Hamiltonian Systems: Geometry Topology Classification offers a unique opportunity to explore important previously unpublished results and acquire generally applicable techniques and tools that enable you to work with a broad class of integrable systems.
Book
This book is devoted to the phenomenon of quasi-periodic motion in dynamical systems. Such a motion in the phase space densely fills up an invariant torus. This phenomenon is most familiar from Hamiltonian dynamics. Hamiltonian systems are well known for their use in modelling the dynamics related to frictionless mechanics, including the planetary and lunar motions. In this context the general picture appears to be as follows. On the one hand, Hamiltonian systems occur that are in complete order: these are the integrable systems where all motion is confined to invariant tori. On the other hand, systems exist that are entirely chaotic on each energy level. In between we know systems that, being sufficiently small perturbations of integrable ones, exhibit coexistence of order (invariant tori carrying quasi-periodic dynamics) and chaos (the so called stochastic layers). The Kolmogorov-Arnol'd-Moser (KAM) theory on quasi-periodic motions tells us that the occurrence of such motions is open within the class of all Hamiltonian systems: in other words, it is a phenomenon persistent under small Hamiltonian perturbations. Moreover, generally, for any such system the union of quasi-periodic tori in the phase space is a nowhere dense set of positive Lebesgue measure, a so called Cantor family. This fact implies that open classes of Hamiltonian systems exist that are not ergodic. The main aim of the book is to study the changes in this picture when other classes of systems - or contexts - are considered.
Article
This book gives a uniquely complete description of the geometry of the energy momentum mapping of five classical integrable systems: the 2-dimensional harmonic oscillator, the geodesic flow on the 3-sphere, the Euler top, the spherical pendulum and the Lagrange top. It presents for the first time in book form a general theory of symmetry reduction which allows one to reduce the symmetries in the spherical pendulum and the Lagrange top. Also the monodromy obstruction to the existence of global action angle coordinates is calculated for the spherical pendulum and the Lagrange top. The book addresses professional mathematicians and graduate students and can be used as a textbook on advanced classical mechanics or global analysis.
Article
About 6 years ago, semitoric systems were classified by Pelayo & Vu Ngoc by means of five invariants. Standard examples are the coupled spin oscillator on $\mathbb{S}^2 \times \mathbb{R}^2$ and coupled angular momenta on $\mathbb{S}^2 \times \mathbb{S}^2$, both having exactly one focus-focus singularity. But so far there were no explicit examples with more than one focus-focus singularity. This paper introduces a 6-parameter family of integrable systems on $\mathbb{S}^2 \times \mathbb{S}^2$ and proves that, for certain ranges of the parameters, it is a compact semitoric system with precisely two focus-focus singularities. Since the twisting index (one of the semitoric invariants) is related to the relationship between different focus-focus points, this paper provides systems for the future study of the - so far unexplored - twisting index.
Book
This monograph presents a comprehensive coverage of three-dimensional topology, as well as exploring some of its frontiers. Many important applied problems of mechanics and theoretical physics can be reduced to algorithmic problems of three-dimensional topology, which can then be solved using computers. Although much progress in this field has been made in recent years, these results have not been readily accessible to a wider audience up to now. This book is based on courses the authors have given over several years, and summarises the most outstanding achievements of modern computer topology. Audience: This book will be of interest to graduate students and researchers whose work involves such diverse disciplines as physics, mathematics, computer programmes for spline theory and its applications, geometrical modelling, geometry, and topology. The illustrations by A.T. Fomenko, drawn especially for this work, add great value and extra appeal.
Article
We consider Hamiltonian systems on (T*ℝ², dq ∧ dp) defined by a Hamiltonian function H of the “classical” form H = p²/2 + V(q). A reasonable decay assumption V(q) → 0, ‖q‖ → ∞, allows one to compare a given distribution of initial conditions at t = −∞ with their final distribution at t = +∞. To describe this Knauf introduced a topological invariant deg(E), which, for a nontrapping energy E > 0, is given by the degree of the scattering map. For rotationally symmetric potentials V(q) = W(‖q‖), scattering monodromy has been introduced independently as another topological invariant. In the present paper we demonstrate that, in the rotationally symmetric case, Knauf’s degree deg(E) and scattering monodromy are related to one another. Specifically, we show that scattering monodromy is given by the jump of the degree deg(E), which appears when the nontrapping energy E goes from low to high values.
Article
The monodromy of torus bundles associated to completely integrable systems can be computed using geometric techniques (constructing homology cycles) or analytic arguments (computing discontinuities of abelian integrals). In this article we give a general approach to the computation of monodromy that resembles the analytical one, reducing the problem to the computation of residues of polar 1-forms. We apply our technique to three celebrated examples of systems with monodromy (the champagne bottle, the spherical pendulum, the hydrogen atom) and to the case of non degenerate focus-focus singularities, re-obtaining the classical results. An advantage of this approach is that the residue-like formula can be shown to be local in a neighborhood of a singularity, hence allowing the definition of monodromy also in the case of non-compact fibers. This idea has been introduced in the literature under the name of scattering monodromy. We prove the coincidence of the two definitions with the monodromy of an appropriately chosen compactification.