We prove that for a Hamiltonian system on a cotangent bundle that is
Liouville-integrable and has monodromy the vector of Maslov indices is an
eigenvector of the monodromy matrix with eigenvalue 1. As a corollary the
resulting restrictions on the monodromy matrix are derived.Comment: 6 page

Arnol'd V I

Bolsinov A V Dullin H R Veselov A P

Cassels J W S

Child M S

Cushman R H

Duistermaat J J

Foxman J A Robbins J M

G Tanner

H R Dullin

H Waalkens

J M Robbins

Maslov V P

Matveev V S

S C Creagh

Trofimov V V

English

arXiv

We prove that for a Hamiltonian system on a cotangent bundle that is Liouville-integrable and has monodromy the vector of Maslov indices is an eigenvector of the monodromy matrix with eigenvalue 1. As a corollary, the resulting restrictions on the monodromy matrix are derived.We prove that for a Hamiltonian system on a cotangent bundle that is Liouville-integrable and has monodromy the vector of Maslov indices is an eigenvector of the monodromy matrix with eigenvalue 1. As a corollary, the resulting restrictions on the monodromy matrix are derived

Dullin, HR

Robbins, JM

Waalkens, H

Creagh, SC

Tanner, G

Explore Bristol Research

Maslov indices and monodromy

Dullin, H.R.

Robbins, J.M.

Waalkens, H.

Creagh, S.C.

Tanner, G.

University of Groningen Research Database

  
 University of Groningen
Maslov indices and monodromy
Dullin, H.R.; Robbins, J.M.; Waalkens, Holger; Creagh, S.C.; Tanner, G.
Published in:
Journal of Physics A: Mathematical and Theoretical
DOI:
10.1088/0305-4470/38/24/L02
IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from
it. Please check the document version below.
Document Version
Publisher's PDF, also known as Version of record
Publication date:
2005
Link to publication in University of Groningen/UMCG research database
Citation for published version (APA):
Dullin, H. R., Robbins, J. M., Waalkens, H., Creagh, S. C., & Tanner, G. (2005). Maslov indices and
monodromy. Journal of Physics A: Mathematical and Theoretical, 38, L443-L447. DOI: 10.1088/0305-
4470/38/24/L02
Copyright
Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the
author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).
Take-down policy
If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately
and investigate your claim.
Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the
number of authors shown on this cover page is limited to 10 maximum.
Download date: 10-02-2018
INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL
J. Phys. A: Math. Gen. 38 (2005) L443–L447 doi:10.1088/0305-4470/38/24/L02
LETTER TO THE EDITOR
Maslov indices and monodromy
H R Dullin1, J M Robbins2, H Waalkens2, S C Creagh3 and G Tanner3
1 Department of Mathematical Sciences, Loughborough University,
Loughborough LE11 3TU, UK
2 School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
3 School of Mathematical Sciences, University of Nottingham, University Park,
Nottingham NG7 2RD, UK
E-mail: H.R.Dullin@lboro.ac.uk
Received 22 April 2005
Published 1 June 2005
Online at stacks.iop.org/JPhysA/38/L443
Abstract
We prove that for a Hamiltonian system on a cotangent bundle that is Liouville-
integrable and has monodromy the vector of Maslov indices is an eigenvector
of the monodromy matrix with eigenvalue 1. As a corollary, the resulting
restrictions on the monodromy matrix are derived.
PACS numbers: 02.30.Ik, 02.40.Ma, 03.65.Vf, 45.20.Jj
Mathematics Subject Classification: 37J35, 81S10, 53D12
1. Introduction
The Liouville–Arnold theorem describes the local structure of an integrable system: for regular
values of the energy–momentum map F : T ∗M → Rn, the preimage of a regular value is
an n-torus (or a union of disconnected n-tori, but for simplicity we assume there is just one),
and there exist action-angle variables in a neighbourhood of this torus. Thus, locally phase
space has the structure of a trivial n-torus-bundle over an open neighbourhood of a regular
value in the image of F. Duistermaat [7] pointed out that globally the torus-bundle over the
regular values of F may be nontrivial. This phenomenon is called monodromy. As a result,
there may not exist global action-angle variables. In 2 degrees of freedom monodromy is well
understood [13, 19]. It is a common phenomenon because it occurs in a neighbourhood of an
equilibrium of focus–focus type. In 3 degrees of freedom now also many examples [17, 18,
8, 3] are known.
Quantization of a classical system with monodromy leads to quantum monodromy
[11, 16, 5, 14, 9]. The fact that the classical actions cannot be globally defined implies
that the quantum numbers suffer the same problem.
The Maslov index is not only interesting for semiclassical quantization, but also in
classical mechanics it is an invariant object defined for paths on Lagrangian submanifolds,
0305-4470/05/240443+05$30.00 © 2005 IOP Publishing Ltd Printed in the UK L443
L444 Letter to the Editor
e.g. on invariant tori, see [1, 12, 2]. Recently, it has been shown that the Maslov index is
related to the singular points of the energy–momentum map [10].
In this letter, we are going to show that if the vector of Maslov indices is non-zero, then
it is an eigenvector of the monodromy matrix with eigenvalue 1. This has some interesting
consequences for the structure of admissible monodromy matrices. Since the Maslov index
is only defined on cotangent bundles our results are only valid when the phase space is a
symplectic manifold of the form T ∗M .
2. Maslov indices
Let C be a closed curve in the set of regular values of the energy–momentum map. We take
C to be parametrized by 0  s  1. Let Ts denote the corresponding one-parameter family
of n-tori in phase space. Fix a basis of cycles γ0 for T0. By continuation this defines a basis
of cycles γs for every s. The curve C has monodromy when γ1 = Mγ0 for M ∈ SL(n,Z)
is nontrivial. More precisely, monodromy is a nontrivial automorphism of the first homology
group, and it implies that the preimage of C under F is a nontrivial n-torus-bundle over C. The
basis γs determines actions Is and Maslov indices µs on Ts . In fact, the Maslov indices are
independent of s, as they depend continuously on s and are integer-valued [15]. Let us denote
their common value by µ. Our main result is the following simple observation:
Theorem 1. If the vector of Maslov indices µ is not equal to zero, then µ is an eigenvector
of the monodromy matrix M with eigenvalue 1.
Proof. We have that µ1 = Mµ0 (just as I1 = MI0), since in general a change of basis cycles
γ ′ = Tγ , where T ∈ SL(n,Z), induces the transformation of Maslov indices µ′ = Tµ (and
the transformation of actions I ′ = TI ). Since µs = µ for all s, µ1 = Mµ0, i.e.
Mµ = µ. 
We remark that the Maslov indices µ, the actions I and the monodromy matrix M depend
on the initial choice of basis γ0. Under a change of basis γ ′0 = Tγ0, where T ∈ SL(n,Z), we
have that µ′ = Tµ and M′ = TMT−1.
3. Monodromy matrices
From theorem 1, we immediately obtain the well-known result [13, 19] about the structure of
monodromy matrices in 2 degrees of freedom:
Corollary 2. For n = 2 degrees of freedom and a loop C with µ = 0 there exists a basis of
cycles such that the monodromy matrix of C has the form
M =
(
1 m
0 1
)
.
Proof. Since M ∈ SL(2,Z) the eigenvalues λ1, λ2 must satisfy λ1λ2 = 1. But one eigenvalue
must be 1 by theorem 1, hence λ1 = λ2 = 1. Finally, a matrix in SL(2,Z) with a single
eigenvalue equal to 1 is conjugate to the stated form by some matrix from SL(2,Z). The
Maslov index in this basis is µ = (µ1, 0). 
Note that this does not give a complete classification of monodromy matrices on cotangent
bundles because we have assumed that µ = 0. When µ = 0, corollary 2 is quite strong because
Letter to the Editor L445
no assumption is needed on the type of singularity that is encircled by C, in particular the
usual non-degeneracy condition is not needed.
Corollary 2 is a special case of the simple general
Lemma 3. Suppose M ∈ SL(n,Z) has eigenvalue ±1. Then there exists T ∈ SL(n,Z) such
that M′ = TMT−1 has first column equal to ±e1 = (±1, 0, . . . , 0)t .
Proof. Let u denote an eigenvector of M with eigenvalue ±1, chosen so that its components
are coprime integers. Then one can construct a matrix S ∈ SL(n,Z) whose first column is u
(see, e.g., [4]). Let T = S−1 and M′ = TMT−1. It is easy to check that e1 is an eigenvector
of M′ with eigenvalue ±1, so that M′ has first column equal to ±e1. 
Using lemma 3 and again the fact that det M = 1 and λ1 = 1, we can obtain the
classification of monodromy matrices (for non-zero Maslov index) in n = 3 degrees of
freedom:
Corollary 4. For n = 3 degrees of freedom and a loop C with µ = 0 there exists a basis of
cycles such that the monodromy matrix M of C has one of the following forms:
1 ∗ ∗0 1 ∗
0 0 1

 ,

1 ∗ ∗0 −1 ∗
0 0 −1

 ,
(
1 ∗
0 B
)
, (1)
where B ∈ SL(2,Z) has irrational eigenvalues and * denotes integers.
Proof. The eigenvalue 1 can appear with algebraic multiplicity ma = 1 or ma = 3 only;
ma = 2 is impossible because det M = λ1λ2λ3 = 1.4 The case ma = 3 corresponds to the
first form above. When ma = 1, the remaining eigenvalues are either both −1, corresponding
to the second form, or they are irrational, corresponding to the third. Other combinations
of eigenvalues are not possible, because rational eigenvalues of matrices in SL(n,Z) are
necessarily equal to ±1.
If the eigenvalues are all ±1 (corresponding to the first two forms), the matrices can be
made upper triangular by applying lemma 3 recursively, using a transformation of the form
Tn =
(
1 ∗
0 Tn−1
)
.
Matrices with two irrational eigenvalues cannot be made upper triangular in SL(n,Z) (as the
diagonal elements of a triangular matrix are its eigenvalues). 
It is interesting to consider how the entries denoted * in (1) can be normalized. In the
first form, the eigenvalue 1 has geometric multiplicity mg equal to 1 or 2 (i.e., there are either
one or two independent eigenvectors with eigenvalue 1). The normal form for mg = 2 has
been computed in [17]. The result is that only a single non-zero element remains above the
diagonal. Essentially this means that when mg = 2, the matrix can be block-diagonalized
in SL(3,Z). In the remaining cases in (1) a block-diagonal form is in general not possible:
conjugating a block triangular matrix with a block triangular matrix gives(
1 −dD−1
0 D−1
)(
1 a
0 A
)(
1 d
0 D
)
=
(
1 (a − dD−1(A − 1))D
0 D−1AD
)
.
Setting the upper right element of the right-hand side to zero and solving for d involves the
inverse of A − 1 which is in general not an integer matrix. Using a more general transformation
4 For general n the multiplicity cannot be n − 1.
L446 Letter to the Editor
leads to the same condition. Thus for general a and A, the monodromy matrix cannot be block-
diagonalized. However, e.g., for the special matrix A = (2 11 1) (the ‘cat map’), it is always
possible since det(A − 1) = −1. If A − 1 is singular (corresponding to the first case in
corollary 4), the resulting Diophantine equation may or may not have a solution.
Our results need the condition µ = 0. It may be possible to show that µ = 0 necessarily
holds for certain configuration spaces M. We suspect, for example, that this is the case when
M = Rn, although we have not been able to prove this. Our result would then give the
complete classification of monodromy matrices on T ∗Rn. In particular, the construction
of arbitrary monodromy matrices given in [6] would be impossible on these cotangent
bundles.
In 3 degrees of freedom, the known examples of monodromy are either of the first form
with mg = 2 [17, 18] or of the last form and block-diagonal. The last form of M is realized
for geodesic flows on Sol-manifolds, where an arbitrary hyperbolic B ∈ SL(2,Z) may
appear [3].
The main implication of the above is that when ma = 3 and mg = 2 there are always two
invariant actions, i.e. actions that do not change globally along the path C. Obviously, there
is always one invariant action, namely the one corresponding to the eigenvector e1, and when
mg = 1 it is the only one. With eigenvalues −1 there is at most one invariant action, but
another action is invariant when C is traversed twice. Hence, on a covering space this may
reduce to ma = 3 and mg = 2. It would be very interesting to find an example of this type.
Acknowledgments
HRD would like to thank AP Veselov and VS Matveev for helpful discussions. This work
was supported by the EU Research Training Network ‘Mechanics and Symmetry in Europe’
(MASIE) (HPRN CT-2000-00113).
References
[1] Arnol’d V I 1967 On a characteristic class entering into conditions of quantization Funct. Anal. Appl. 1 1–13
[2] Arnol’d V I and Givental’ A B 2001 Symplectic geometry Dynamical Systems IV (Encyclopaedia Math. Sci.,
vol 4) (Berlin: Springer) pp 1–138
[3] Bolsinov A V, Dullin H R and Veselov A P 2005 Sol-manifolds: spectrum and quantum monodromy
http://arXiv.org/abs/math-ph/0503046
[4] Cassels J W S 1997 An introduction to the geometry of numbers Classics in Mathematics (Berlin: Springer)
(Corrected reprint of the 1971 edition)
[5] Child M S 1998 Quantum states in a champagne bottle J. Phys. A: Math. Gen. 31 657–70
[6] Cushman R and Vu˜ Ngo
.
c S 2002 Sign of the monodromy for Liouville integrable systems Ann. Henri Poincare´
3 883–94
[7] Duistermaat J J 1980 On global action-angle coordinates Commun. Pure Appl. Math. 33 687–706
[8] Dullin H R, Giacobbe A and Cushman R 2004 Monodromy in the resonant swing spring Physica D 190 15–37
[9] Efstathiou K, Cushman R H and Sadovskiı´ D A 2004 Hamiltonian Hopf bifurcation of the hydrogen atom in
crossed fields Physica D 194 250–74
[10] Foxman J A and Robbins J M 2004 The Maslov index and nondegenerate singularities of integrable systems
http://arxiv.org/abs/math-ph/0411018
[11] Cushman R H and Duistermaat J J 1988 The quantum mechanical spherical pendulum Bull. Am. Math. Soc. 19
475–9
[12] Maslov V P and Fedoriuk M 1981 Semiclassical Approximation in Quantum Mechanics (Dordrecht: Reidel)
[13] Matveev V S 1996 Integrable Hamiltonian systems with two degrees of freedom. Topological structure of
saturated neighborhoods of saddle–saddle and focus–focus types Matem. Sbornik 187 29–58
[14] Sadovskiı´ D A and Zˆhilinskiı´ B I 1999 Monodromy, diabolic points, and angular momentum coupling Phys.
Lett. A 256 235–44
Letter to the Editor L447
[15] Trofimov V V 1995 Generalized Maslov classes on the path space of a symplectic manifold Proc. Steklov. Inst.
Math. 205 157–79
[16] Vu˜ Ngo
.
c S 1999 Quantum monodromy in integrable systems Commun. Math. Phys. 203 465–79
[17] Waalkens H and Dullin H R 2002 Quantum monodromy in prolate ellipsoidal billiards Ann. Phys. 295 81–112
[18] Waalkens H, Dullin H R and Richter P H 2004 The problem of two fixed centers: bifurcations, actions,
monodromy Physica D 196 265–310
[19] Zung N T 1997 A note on focus–focus singularities Differ. Geom. Appl. 7 123–30


University of Groningen

   University of GroningenMaslov indices and monodromyDullin, H.R.; Robbins, J.M.; Waalkens, H.; Creagh, S.C.; Tanner, G.Published in:Journal of Physics A: Mathematical and TheoreticalDOI:10.1088/0305-4470/38/24/L02IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2005Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Dullin, H. R., Robbins, J. M., Waalkens, H., Creagh, S. C., & Tanner, G. (2005). Maslov indices andmonodromy. Journal of Physics A: Mathematical and Theoretical, 38, L443-L447.https://doi.org/10.1088/0305-4470/38/24/L02CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 12-11-2019INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERALJ. Phys. A: Math. Gen. 38 (2005) L443–L447 doi:10.1088/0305-4470/38/24/L02LETTER TO THE EDITORMaslov indices and monodromyH R Dullin1, J M Robbins2, H Waalkens2, S C Creagh3 and G Tanner31 Department of Mathematical Sciences, Loughborough University,Loughborough LE11 3TU, UK2 School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK3 School of Mathematical Sciences, University of Nottingham, University Park,Nottingham NG7 2RD, UKE-mail: H.R.Dullin@lboro.ac.ukReceived 22 April 2005Published 1 June 2005Online at stacks.iop.org/JPhysA/38/L443AbstractWe prove that for a Hamiltonian system on a cotangent bundle that is Liouville-integrable and has monodromy the vector of Maslov indices is an eigenvectorof the monodromy matrix with eigenvalue 1. As a corollary, the resultingrestrictions on the monodromy matrix are derived.PACS numbers: 02.30.Ik, 02.40.Ma, 03.65.Vf, 45.20.JjMathematics Subject Classification: 37J35, 81S10, 53D121. IntroductionThe Liouville–Arnold theorem describes the local structure of an integrable system: for regularvalues of the energy–momentum map F : T ∗M → Rn, the preimage of a regular value isan n-torus (or a union of disconnected n-tori, but for simplicity we assume there is just one),and there exist action-angle variables in a neighbourhood of this torus. Thus, locally phasespace has the structure of a trivial n-torus-bundle over an open neighbourhood of a regularvalue in the image of F. Duistermaat [7] pointed out that globally the torus-bundle over theregular values of F may be nontrivial. This phenomenon is called monodromy. As a result,there may not exist global action-angle variables. In 2 degrees of freedom monodromy is wellunderstood [13, 19]. It is a common phenomenon because it occurs in a neighbourhood of anequilibrium of focus–focus type. In 3 degrees of freedom now also many examples [17, 18,8, 3] are known.Quantization of a classical system with monodromy leads to quantum monodromy[11, 16, 5, 14, 9]. The fact that the classical actions cannot be globally defined impliesthat the quantum numbers suffer the same problem.The Maslov index is not only interesting for semiclassical quantization, but also inclassical mechanics it is an invariant object defined for paths on Lagrangian submanifolds,0305-4470/05/240443+05$30.00 © 2005 IOP Publishing Ltd Printed in the UK L443L444 Letter to the Editore.g. on invariant tori, see [1, 12, 2]. Recently, it has been shown that the Maslov index isrelated to the singular points of the energy–momentum map [10].In this letter, we are going to show that if the vector of Maslov indices is non-zero, thenit is an eigenvector of the monodromy matrix with eigenvalue 1. This has some interestingconsequences for the structure of admissible monodromy matrices. Since the Maslov indexis only defined on cotangent bundles our results are only valid when the phase space is asymplectic manifold of the form T ∗M .2. Maslov indicesLet C be a closed curve in the set of regular values of the energy–momentum map. We takeC to be parametrized by 0  s  1. Let Ts denote the corresponding one-parameter familyof n-tori in phase space. Fix a basis of cycles γ0 for T0. By continuation this defines a basisof cycles γs for every s. The curve C has monodromy when γ1 = Mγ0 for M ∈ SL(n,Z)is nontrivial. More precisely, monodromy is a nontrivial automorphism of the first homologygroup, and it implies that the preimage of C under F is a nontrivial n-torus-bundle over C. Thebasis γs determines actions Is and Maslov indices µs on Ts . In fact, the Maslov indices areindependent of s, as they depend continuously on s and are integer-valued [15]. Let us denotetheir common value by µ. Our main result is the following simple observation:Theorem 1. If the vector of Maslov indices µ is not equal to zero, then µ is an eigenvectorof the monodromy matrix M with eigenvalue 1.Proof. We have that µ1 = Mµ0 (just as I1 = MI0), since in general a change of basis cyclesγ ′ = Tγ , where T ∈ SL(n,Z), induces the transformation of Maslov indices µ′ = Tµ (andthe transformation of actions I ′ = TI ). Since µs = µ for all s, µ1 = Mµ0, i.e.Mµ = µ. We remark that the Maslov indices µ, the actions I and the monodromy matrix M dependon the initial choice of basis γ0. Under a change of basis γ ′0 = Tγ0, where T ∈ SL(n,Z), wehave that µ′ = Tµ and M′ = TMT−1.3. Monodromy matricesFrom theorem 1, we immediately obtain the well-known result [13, 19] about the structure ofmonodromy matrices in 2 degrees of freedom:Corollary 2. For n = 2 degrees of freedom and a loop C with µ = 0 there exists a basis ofcycles such that the monodromy matrix of C has the formM =(1 m0 1).Proof. Since M ∈ SL(2,Z) the eigenvalues λ1, λ2 must satisfy λ1λ2 = 1. But one eigenvaluemust be 1 by theorem 1, hence λ1 = λ2 = 1. Finally, a matrix in SL(2,Z) with a singleeigenvalue equal to 1 is conjugate to the stated form by some matrix from SL(2,Z). TheMaslov index in this basis is µ = (µ1, 0). Note that this does not give a complete classification of monodromy matrices on cotangentbundles because we have assumed that µ = 0. When µ = 0, corollary 2 is quite strong becauseLetter to the Editor L445no assumption is needed on the type of singularity that is encircled by C, in particular theusual non-degeneracy condition is not needed.Corollary 2 is a special case of the simple generalLemma 3. Suppose M ∈ SL(n,Z) has eigenvalue ±1. Then there exists T ∈ SL(n,Z) suchthat M′ = TMT−1 has first column equal to ±e1 = (±1, 0, . . . , 0)t .Proof. Let u denote an eigenvector of M with eigenvalue ±1, chosen so that its componentsare coprime integers. Then one can construct a matrix S ∈ SL(n,Z) whose first column is u(see, e.g., [4]). Let T = S−1 and M′ = TMT−1. It is easy to check that e1 is an eigenvectorof M′ with eigenvalue ±1, so that M′ has first column equal to ±e1. Using lemma 3 and again the fact that det M = 1 and λ1 = 1, we can obtain theclassification of monodromy matrices (for non-zero Maslov index) in n = 3 degrees offreedom:Corollary 4. For n = 3 degrees of freedom and a loop C with µ = 0 there exists a basis ofcycles such that the monodromy matrix M of C has one of the following forms:1 ∗ ∗0 1 ∗0 0 1 ,1 ∗ ∗0 −1 ∗0 0 −1 ,(1 ∗0 B), (1)where B ∈ SL(2,Z) has irrational eigenvalues and * denotes integers.Proof. The eigenvalue 1 can appear with algebraic multiplicity ma = 1 or ma = 3 only;ma = 2 is impossible because det M = λ1λ2λ3 = 1.4 The case ma = 3 corresponds to thefirst form above. When ma = 1, the remaining eigenvalues are either both −1, correspondingto the second form, or they are irrational, corresponding to the third. Other combinationsof eigenvalues are not possible, because rational eigenvalues of matrices in SL(n,Z) arenecessarily equal to ±1.If the eigenvalues are all ±1 (corresponding to the first two forms), the matrices can bemade upper triangular by applying lemma 3 recursively, using a transformation of the formTn =(1 ∗0 Tn−1).Matrices with two irrational eigenvalues cannot be made upper triangular in SL(n,Z) (as thediagonal elements of a triangular matrix are its eigenvalues). It is interesting to consider how the entries denoted * in (1) can be normalized. In thefirst form, the eigenvalue 1 has geometric multiplicity mg equal to 1 or 2 (i.e., there are eitherone or two independent eigenvectors with eigenvalue 1). The normal form for mg = 2 hasbeen computed in [17]. The result is that only a single non-zero element remains above thediagonal. Essentially this means that when mg = 2, the matrix can be block-diagonalizedin SL(3,Z). In the remaining cases in (1) a block-diagonal form is in general not possible:conjugating a block triangular matrix with a block triangular matrix gives(1 −dD−10 D−1)(1 a0 A)(1 d0 D)=(1 (a − dD−1(A − 1))D0 D−1AD).Setting the upper right element of the right-hand side to zero and solving for d involves theinverse of A − 1 which is in general not an integer matrix. Using a more general transformation4 For general n the multiplicity cannot be n − 1.L446 Letter to the Editorleads to the same condition. Thus for general a and A, the monodromy matrix cannot be block-diagonalized. However, e.g., for the special matrix A = (2 11 1) (the ‘cat map’), it is alwayspossible since det(A − 1) = −1. If A − 1 is singular (corresponding to the first case incorollary 4), the resulting Diophantine equation may or may not have a solution.Our results need the condition µ = 0. It may be possible to show that µ = 0 necessarilyholds for certain configuration spaces M. We suspect, for example, that this is the case whenM = Rn, although we have not been able to prove this. Our result would then give thecomplete classification of monodromy matrices on T ∗Rn. In particular, the constructionof arbitrary monodromy matrices given in [6] would be impossible on these cotangentbundles.In 3 degrees of freedom, the known examples of monodromy are either of the first formwith mg = 2 [17, 18] or of the last form and block-diagonal. The last form of M is realizedfor geodesic flows on Sol-manifolds, where an arbitrary hyperbolic B ∈ SL(2,Z) mayappear [3].The main implication of the above is that when ma = 3 and mg = 2 there are always twoinvariant actions, i.e. actions that do not change globally along the path C. Obviously, thereis always one invariant action, namely the one corresponding to the eigenvector e1, and whenmg = 1 it is the only one. With eigenvalues −1 there is at most one invariant action, butanother action is invariant when C is traversed twice. Hence, on a covering space this mayreduce to ma = 3 and mg = 2. It would be very interesting to find an example of this type.AcknowledgmentsHRD would like to thank AP Veselov and VS Matveev for helpful discussions. This workwas supported by the EU Research Training Network ‘Mechanics and Symmetry in Europe’(MASIE) (HPRN CT-2000-00113).References[1] Arnol’d V I 1967 On a characteristic class entering into conditions of quantization Funct. Anal. Appl. 1 1–13[2] Arnol’d V I and Givental’ A B 2001 Symplectic geometry Dynamical Systems IV (Encyclopaedia Math. Sci.,vol 4) (Berlin: Springer) pp 1–138[3] Bolsinov A V, Dullin H R and Veselov A P 2005 Sol-manifolds: spectrum and quantum monodromyhttp://arXiv.org/abs/math-ph/0503046[4] Cassels J W S 1997 An introduction to the geometry of numbers Classics in Mathematics (Berlin: Springer)(Corrected reprint of the 1971 edition)[5] Child M S 1998 Quantum states in a champagne bottle J. Phys. A: Math. Gen. 31 657–70[6] Cushman R and Vu˜ Ngo.c S 2002 Sign of the monodromy for Liouville integrable systems Ann. Henri Poincare´3 883–94[7] Duistermaat J J 1980 On global action-angle coordinates Commun. Pure Appl. Math. 33 687–706[8] Dullin H R, Giacobbe A and Cushman R 2004 Monodromy in the resonant swing spring Physica D 190 15–37[9] Efstathiou K, Cushman R H and Sadovskiı´ D A 2004 Hamiltonian Hopf bifurcation of the hydrogen atom incrossed fields Physica D 194 250–74[10] Foxman J A and Robbins J M 2004 The Maslov index and nondegenerate singularities of integrable systemshttp://arxiv.org/abs/math-ph/0411018[11] Cushman R H and Duistermaat J J 1988 The quantum mechanical spherical pendulum Bull. Am. Math. Soc. 19475–9[12] Maslov V P and Fedoriuk M 1981 Semiclassical Approximation in Quantum Mechanics (Dordrecht: Reidel)[13] Matveev V S 1996 Integrable Hamiltonian systems with two degrees of freedom. Topological structure ofsaturated neighborhoods of saddle–saddle and focus–focus types Matem. Sbornik 187 29–58[14] Sadovskiı´ D A and Zˆhilinskiı´ B I 1999 Monodromy, diabolic points, and angular momentum coupling Phys.Lett. A 256 235–44Letter to the Editor L447[15] Trofimov V V 1995 Generalized Maslov classes on the path space of a symplectic manifold Proc. Steklov. Inst.Math. 205 157–79[16] Vu˜ Ngo.c S 1999 Quantum monodromy in integrable systems Commun. Math. Phys. 203 465–79[17] Waalkens H and Dullin H R 2002 Quantum monodromy in prolate ellipsoidal billiards Ann. Phys. 295 81–112[18] Waalkens H, Dullin H R and Richter P H 2004 The problem of two fixed centers: bifurcations, actions,monodromy Physica D 196 265–310[19] Zung N T 1997 A note on focus–focus singularities Differ. Geom. Appl. 7 123–30

We prove that for a Hamiltonian system on a cotangent bundle that is Liouville-integrable and has monodromy the vector of Maslov indices is an eigenvector of the monodromy matrix with eigenvalue 1. As a corollary the resulting restrictions on the monodromy matrix are derived

Holger R. Dullin (7159922)

J.M. Robbins (7160249)

Holger Waalkens (7160153)

S.C. Creagh (7160252)

G. Tanner (7160255)

Loughborough University Institutional Repository

Maslov Indices and MonodromyHR DullinDepartment of Mathematical SciencesLoughborough UniversityLoughborough LE11 3TUH.R.Dullin@lboro.ac.ukJM Robbins and H WaalkensSchool of MathematicsUniversity of BristolUniversity WalkBristol BS8 1TWSC Creagh and G TannerSchool of Mathematical SciencesUniversity of NottinghamUniversity ParkNottingham NG7 2RDApril 2005AbstractWe prove that for a Hamiltonian system on a cotangent bundle that isLiouville-integrable and has monodromy the vector of Maslov indices is aneigenvector of the monodromy matrix with eigenvalue 1. As a corollary theresulting restrictions on the monodromy matrix are derived.1 IntroductionThe Liouville-Arnold theorem describes the local structure of an integrablesystem: for regular values of the energy-momentum map F : T ∗M → Rn, thepreimage of a regular value is an n-torus (or a union of disconnected n-tori,but for simplicity we assume there is just one), and there exist action-anglevariables in a neighbourhood of this torus. Thus locally phase space has thestructure of a trivial n-torus-bundle over an open neighbourhood of a regularvalue in the image of F . Duistermaat [7] pointed out that globally the torus-bundle over the regular values of F may be non-trivial. This phenomenon iscalled monodromy. As a result there may not exist global action-angle vari-ables. In two degrees of freedom monodromy is well understood [13, 19]. It isa common phenomenon because it occurs in a neighbourhood of an equilib-rium of focus-focus type. In three degrees of freedom now also many examples[17, 18, 8, 3] are known.Quantisation of a classical system with monodromy leads to quantum mon-odromy [11, 16, 5, 14, 9]. The fact that the classical actions cannot be globallydefined implies that the quantum numbers suffer the same problem.1The Maslov index is not only interesting for semiclassical quantisation,but also in classical mechanics it is an invariant object defined for paths onLagrangian submanifolds, e.g. on invariant tori, see [1, 12, 2]. Recently ithas been shown that the Maslov index is related to the singular points of theenergy-momentum map [10].In this letter we are going to show that if the vector of Maslov indices isnon-zero, then it is an eigenvector of the monodromy matrix with eigenvalue1. This has some interesting consequences for the structure of admissible mon-odromy matrices. Since the Maslov index is only defined on cotangent bundlesour results are only valid when the phase space is a symplectic manifold ofthe form T ∗M .2 Maslov indicesLet C be a closed curve in the set of regular values of the energy-momentummap. We take C to be parameterised by 0 ≤ s ≤ 1. Let Ts denote thecorresponding one-parameter family of n-tori in phase space. Fix a basis ofcycles γ0 for T0. By continuation this defines a basis of cycles γs for every s.The curve C has monodromy when γ1 =Mγ0 for M ∈ SL(n,Z) is nontrivial.More precisely, monodromy is a nontrivial automorphism of the first homologygroup, and it implies that the preimage of C under F is a nontrivial n-torus-bundle over C. The basis γs determines actions Is and Maslov indices µs on Ts.In fact, the Maslov indices are independent of s, as they depend continuouslyon s and are integer-valued [15]. Let us denote their common value by µ. Ourmain result is the following simple observation:Theorem 1. If the vector of Maslov indices µ is not equal to zero, then µ isan eigenvector of the monodromy matrix M with eigenvalue 1.Proof. We have that µ1 =Mµ0 (just as I1 =MI0), since in general a changeof basis cycles γ′ = Tγ, where T ∈ SL(n,Z), induces the transformation ofMaslov indices µ′ = Tµ (and the transformation of actions I ′ = TI). Sinceµs = µ for all s, µ1 =Mµ0, i.e.Mµ = µ .We remark that the Maslov indices µ, the actions I, and the monodromymatrix M depend on the initial choice of basis γ0. Under a change of basisγ′0 = Tγ0, where T ∈ SL(n,Z), we have that µ′ = Tµ and M′ = TMT−1.3 Monodromy matricesFrom Theorem 1 we immediately obtain the well-known result [13, 19] aboutthe structure of monodromy matrices in two degrees of freedom:2Corollary 2. For n = 2 degrees of freedom and a loop C with µ 6= 0 thereexists a basis of cycles such that the monodromy matrix of C has the formM =(1 m0 1).Proof. Since M ∈ SL(2,Z) the eigenvalues λ1, λ2 must satisfy λ1λ2 = 1.But one eigenvalue must be 1 by Theorem 1, hence λ1 = λ2 = 1. Finallya matrix in SL(2,Z) with a single eigenvalue equal to 1 is conjugate to thestated form by some matrix from SL(2,Z). The Maslov index in this basis isµ = (µ1, 0).Notice that this does not give a complete classification of monodromymatrices on cotangent bundles because we have assumed that µ 6= 0. Whenµ 6= 0, Corollary 2 is quite strong because no assumption is needed on the typeof singularity that is encircled by C, in particular the usual non-degeneracycondition is not needed.Corollary 2 is a special case of the simple generalLemma 3. Suppose M ∈ SL(n,Z) has eigenvalue ±1. Then there exists T ∈SL(n,Z) such M′ = TMT−1 has first column equal to ±e1 = (±1, 0, . . . , 0)t.Proof. Let u denote an eigenvector of M with eigenvalue ±1, chosen so thatits components are coprime integers. Then one can construct a matrix S ∈SL(n,Z) whose first column is u (see, e.g., [4]). Let T = S−1 and M′ =TMT−1. It is easy to check that e1 is an eigenvector of M′ with eigenvalue±1, so that M′ has first column equal to ±e1.Using Lemma 3 and again the fact that detM = 1 and λ1 = 1, we canobtain the classification of monodromy matrices (for non-zero Maslov index)in n = 3 degrees of freedom:Corollary 4. For n = 3 degrees of freedom and a loop C with µ 6= 0 thereexists a basis of cycles such that the monodromy matrix M of C has one ofthe following forms:1 ∗ ∗0 1 ∗0 0 1 ,1 ∗ ∗0 −1 ∗0 0 −1 , (1 ∗0 B), (1)where B ∈ SL(2,Z) has irrational eigenvalues and ∗ denotes integers.Proof. The eigenvalue 1 can appear with algebraic multiplicity ma = 1 orma = 3 only; ma = 2 is impossible because detM = λ1λ2λ3 = 1. 1 The casema = 3 corresponds to the first form above. When ma = 1, the remainingeigenvalues are either both −1, corresponding to the second form, or theyare irrational, corresponding to the third. Other combinations of eigenvaluesare not possible, because rational eigenvalues of matrices in SL(n,Z) arenecessarily equal to ±1.1For general n the multiplicity cannot be n− 1.3If the eigenvalues are all ±1 (corresponding to the first two forms), thematrices can be made upper triangular by applying Lemma 3 recursively,using a transformation of the formTn =(1 ∗0 Tn−1).Matrices with two irrational eigenvalues cannot be made upper triangular inSL(n,Z) (as the diagonal elements of a triangular matrix are its eigenvalues).It is interesting to consider how the entries denoted ∗ in (1) can be nor-malised. In the first form the eigenvalue 1 has geometric multiplicity mgequal to 1 or 2 (i.e., there are either one or two independent eigenvectors witheigenvalue 1). The normal form for mg = 2 has been computed in [17]. Theresult is that only a single nonzero element remains above the diagonal. Es-sentially this means that when mg = 2 the matrix can be block-diagonalisedin SL(3,Z). In the remaining cases in (1) a block-diagonal form is in generalnot possible: Conjugating a block triangular matrix with a block triangularmatrix gives(1 −dD−10 D−1)(1 a0 A)(1 d0 D)=(1 (a− dD−1(A− 1))D0 D−1AD).Setting the upper right element of the right hand side to zero and solving for dinvolves the inverse of A− 1 which is in general not an integer matrix. Usinga more general transformation leads to the same condition. Thus for generala and A the monodromy matrix cannot be block-diagonalised. However,e.g., for the special matrix A =(2 11 1)(the ‘cat map’) it is always possiblesince det(A− 1) = −1. If A− 1 is singular (corresponding to the first casein Corollary 4) the resulting Diophantine equation may or may not have asolution.Our results need the condition µ 6= 0. It may be possible to show thatµ 6= 0 necessarily holds for certain configuration spaces M . We suspect, forexample, that this is the case when M = Rn, although we have not beenable to prove this. Our result would then give the complete classification ofmonodromy matrices on T ∗Rn. In particular the construction of arbitrarymonodromy matrices given in [6] would be impossible on these cotangentbundles.In three degrees of freedom, the known examples of monodromy are eitherof the first form with mg = 2 [17, 18] or of the last form and block-diagonal.The last form of M is realised for geodesic flows on Sol-manifolds, where anarbitrary hyperbolic B ∈ SL(2,Z) may appear [3].The main implication of the above is that when ma = 3 and mg = 2 thereare always two invariant actions, i.e. actions that do not change globally alongthe path C. Obviously there is always one invariant action, namely the onecorresponding to the eigenvector e1, and when mg = 1 it is the only one.With eigenvalues −1 there is at most one invariant action, but another actionis invariant when C is traversed twice. Hence on a covering space this may4reduce to ma = 3 and mg = 2. It would be very interesting to find an exampleof this type.AcknowledgementHRD would like to that AP Veselov and VS Matveev for helpful discussions.This work was supported by the EU Research Training Network ”Mechanicsand Symmetry in Europe” (MASIE), (HPRN CT-2000-00113).References[1] V. I. Arnol′d. On a characteristic class entering into conditions of quan-tization. Funkcional Anal. Appl., 1:1–13, 1967.[2] V. I. Arnol′d and A. B. Givental′. Symplectic geometry. In Dynam-ical systems, IV, volume 4 of Encyclopaedia Math. Sci., pages 1–138.Springer, Berlin, 2001.[3] A. V. Bolsinov, H. R. Dullin, and A. P. Veselov. Sol-manifolds: spec-trum and quantum monodromy. http://arXiv.org/abs/math-ph/0503046,2005.[4] J. W. S. Cassels. An introduction to the geometry of numbers. Classicsin Mathematics. Springer-Verlag, Berlin, 1997. Corrected reprint of the1971 edition.[5] M. S. Child. Quantum states in a champagne bottle. J. Phys. A,31(2):657–670, 1998.[6] R. Cushman and Vu˜ Ngoc San. Sign of the monodromy for Liouvilleintegrable systems. Ann. Henri Poincare´, 3(5):883–894, 2002.[7] J. J. Duistermaat. On global action-angle coordinates. Comm. PureAppl. Math., 33:687–706, 1980.[8] H. R. Dullin, A. Giacobbe, and R. Cushman. Monodromy in the resonantswing spring. Physica D, 190:15–37, 2004.[9] K. Efstathiou, R. H. Cushman, and D. A. Sadovski´ı. Hamiltonian Hopfbifurcation of the hydrogen atom in crossed fields. Phys. D, 194(3-4):250–274, 2004.[10] J.A. Foxman and J.M. Robbins. The Maslov index and nondegen-erate singularities of integrable systems. http://arxiv.org/abs/math-ph/0411018, 2004.[11] Cushman R. H. and J. J. Duistermaat. The quantum mechanical spher-ical pendulum. Bull. Amer. Math. Soc., 19:475–479, 1988.[12] V.P. Maslov and M. Fedoriuk. Semiclassical approximation in quantummechanics. Reidel, 1981.[13] V. S. Matveev. Integrable Hamiltonian systems with two degrees of free-dom. Topological structure of saturated neighborhoods of saddle-saddleand focus-focus types. Matem. Sbornik, 187(4):29–58, 1996.5[14] D. A. Sadovski´ı and B. I. Zhˆilinski´ı. Monodromy, diabolic points, andangular momentum coupling. Phys. Lett. A, 256(4):235–244, 1999.[15] V. V. Trofimov. Generalized Maslov classes on the path space of a sym-plectic manifold. Proc. Steklov. Inst. Math., 205:157–179, 1995.[16] San Vu˜ Ngo.c. Quantum monodromy in integrable systems. Comm. Math.Phys., 203(2):465–479, 1999.[17] H. Waalkens and H. R. Dullin. Quantum monodromy in prolate ellip-soidal billiards. Ann. Physics, 295:81–112, 2002.[18] H. Waalkens, H. R. Dullin, and P. H. Richter. The problem of two fixedcenters: bifurcations, actions, monodromy. Physica D, 196:265–310, 2004.[19] Nguyen Tien Zung. A note on focus-focus singularities. DifferentialGeom. Appl., 7(2):123–130, 1997.6

We prove that for a Hamiltonian system on a cotangent bundle that is Liouville-integrable and has monodromy the vector of Maslov indices is an eigenvector of the monodromy matrix with eigenvalue 1. As a corollary, the resulting restrictions on the monodromy matrix are derived

NARCIS 

Crossref

   University of GroningenMaslov indices and monodromyDullin, H.R.; Robbins, J.M.; Waalkens, H.; Creagh, S.C.; Tanner, G.Published in:Journal of Physics A: Mathematical and TheoreticalDOI:10.1088/0305-4470/38/24/L02IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2005Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Dullin, H. R., Robbins, J. M., Waalkens, H., Creagh, S. C., & Tanner, G. (2005). Maslov indices andmonodromy. Journal of Physics A: Mathematical and Theoretical, 38, L443-L447.https://doi.org/10.1088/0305-4470/38/24/L02CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 25-04-2023INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERALJ. Phys. A: Math. Gen. 38 (2005) L443–L447 doi:10.1088/0305-4470/38/24/L02LETTER TO THE EDITORMaslov indices and monodromyH R Dullin1, J M Robbins2, H Waalkens2, S C Creagh3 and G Tanner31 Department of Mathematical Sciences, Loughborough University,Loughborough LE11 3TU, UK2 School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK3 School of Mathematical Sciences, University of Nottingham, University Park,Nottingham NG7 2RD, UKE-mail: H.R.Dullin@lboro.ac.ukReceived 22 April 2005Published 1 June 2005Online at stacks.iop.org/JPhysA/38/L443AbstractWe prove that for a Hamiltonian system on a cotangent bundle that is Liouville-integrable and has monodromy the vector of Maslov indices is an eigenvectorof the monodromy matrix with eigenvalue 1. As a corollary, the resultingrestrictions on the monodromy matrix are derived.PACS numbers: 02.30.Ik, 02.40.Ma, 03.65.Vf, 45.20.JjMathematics Subject Classification: 37J35, 81S10, 53D121. IntroductionThe Liouville–Arnold theorem describes the local structure of an integrable system: for regularvalues of the energy–momentum map F : T ∗M → Rn, the preimage of a regular value isan n-torus (or a union of disconnected n-tori, but for simplicity we assume there is just one),and there exist action-angle variables in a neighbourhood of this torus. Thus, locally phasespace has the structure of a trivial n-torus-bundle over an open neighbourhood of a regularvalue in the image of F. Duistermaat [7] pointed out that globally the torus-bundle over theregular values of F may be nontrivial. This phenomenon is called monodromy. As a result,there may not exist global action-angle variables. In 2 degrees of freedom monodromy is wellunderstood [13, 19]. It is a common phenomenon because it occurs in a neighbourhood of anequilibrium of focus–focus type. In 3 degrees of freedom now also many examples [17, 18,8, 3] are known.Quantization of a classical system with monodromy leads to quantum monodromy[11, 16, 5, 14, 9]. The fact that the classical actions cannot be globally defined impliesthat the quantum numbers suffer the same problem.The Maslov index is not only interesting for semiclassical quantization, but also inclassical mechanics it is an invariant object defined for paths on Lagrangian submanifolds,0305-4470/05/240443+05$30.00 © 2005 IOP Publishing Ltd Printed in the UK L443L444 Letter to the Editore.g. on invariant tori, see [1, 12, 2]. Recently, it has been shown that the Maslov index isrelated to the singular points of the energy–momentum map [10].In this letter, we are going to show that if the vector of Maslov indices is non-zero, thenit is an eigenvector of the monodromy matrix with eigenvalue 1. This has some interestingconsequences for the structure of admissible monodromy matrices. Since the Maslov indexis only defined on cotangent bundles our results are only valid when the phase space is asymplectic manifold of the form T ∗M .2. Maslov indicesLet C be a closed curve in the set of regular values of the energy–momentum map. We takeC to be parametrized by 0  s  1. Let Ts denote the corresponding one-parameter familyof n-tori in phase space. Fix a basis of cycles γ0 for T0. By continuation this defines a basisof cycles γs for every s. The curve C has monodromy when γ1 = Mγ0 for M ∈ SL(n, Z)is nontrivial. More precisely, monodromy is a nontrivial automorphism of the first homologygroup, and it implies that the preimage of C under F is a nontrivial n-torus-bundle over C. Thebasis γs determines actions Is and Maslov indices µs on Ts . In fact, the Maslov indices areindependent of s, as they depend continuously on s and are integer-valued [15]. Let us denotetheir common value by µ. Our main result is the following simple observation:Theorem 1. If the vector of Maslov indices µ is not equal to zero, then µ is an eigenvectorof the monodromy matrix M with eigenvalue 1.Proof. We have that µ1 = Mµ0 (just as I1 = MI0), since in general a change of basis cyclesγ ′ = Tγ , where T ∈ SL(n, Z), induces the transformation of Maslov indices µ′ = Tµ (andthe transformation of actions I ′ = TI ). Since µs = µ for all s, µ1 = Mµ0, i.e.Mµ = µ. We remark that the Maslov indices µ, the actions I and the monodromy matrix M dependon the initial choice of basis γ0. Under a change of basis γ ′0 = Tγ0, where T ∈ SL(n, Z), wehave that µ′ = Tµ and M′ = TMT−1.3. Monodromy matricesFrom theorem 1, we immediately obtain the well-known result [13, 19] about the structure ofmonodromy matrices in 2 degrees of freedom:Corollary 2. For n = 2 degrees of freedom and a loop C with µ = 0 there exists a basis ofcycles such that the monodromy matrix of C has the formM =(1 m0 1).Proof. Since M ∈ SL(2, Z) the eigenvalues λ1, λ2 must satisfy λ1λ2 = 1. But one eigenvaluemust be 1 by theorem 1, hence λ1 = λ2 = 1. Finally, a matrix in SL(2, Z) with a singleeigenvalue equal to 1 is conjugate to the stated form by some matrix from SL(2, Z). TheMaslov index in this basis is µ = (µ1, 0). Note that this does not give a complete classification of monodromy matrices on cotangentbundles because we have assumed that µ = 0. When µ = 0, corollary 2 is quite strong becauseLetter to the Editor L445no assumption is needed on the type of singularity that is encircled by C, in particular theusual non-degeneracy condition is not needed.Corollary 2 is a special case of the simple generalLemma 3. Suppose M ∈ SL(n, Z) has eigenvalue ±1. Then there exists T ∈ SL(n, Z) suchthat M′ = TMT−1 has first column equal to ±e1 = (±1, 0, . . . , 0)t .Proof. Let u denote an eigenvector of M with eigenvalue ±1, chosen so that its componentsare coprime integers. Then one can construct a matrix S ∈ SL(n, Z) whose first column is u(see, e.g., [4]). Let T = S−1 and M′ = TMT−1. It is easy to check that e1 is an eigenvectorof M′ with eigenvalue ±1, so that M′ has first column equal to ±e1. Using lemma 3 and again the fact that det M = 1 and λ1 = 1, we can obtain theclassification of monodromy matrices (for non-zero Maslov index) in n = 3 degrees offreedom:Corollary 4. For n = 3 degrees of freedom and a loop C with µ = 0 there exists a basis ofcycles such that the monodromy matrix M of C has one of the following forms:1 ∗ ∗0 1 ∗0 0 1 ,1 ∗ ∗0 −1 ∗0 0 −1 ,(1 ∗0 B), (1)where B ∈ SL(2, Z) has irrational eigenvalues and * denotes integers.Proof. The eigenvalue 1 can appear with algebraic multiplicity ma = 1 or ma = 3 only;ma = 2 is impossible because det M = λ1λ2λ3 = 1.4 The case ma = 3 corresponds to thefirst form above. When ma = 1, the remaining eigenvalues are either both −1, correspondingto the second form, or they are irrational, corresponding to the third. Other combinationsof eigenvalues are not possible, because rational eigenvalues of matrices in SL(n, Z) arenecessarily equal to ±1.If the eigenvalues are all ±1 (corresponding to the first two forms), the matrices can bemade upper triangular by applying lemma 3 recursively, using a transformation of the formTn =(1 ∗0 Tn−1).Matrices with two irrational eigenvalues cannot be made upper triangular in SL(n, Z) (as thediagonal elements of a triangular matrix are its eigenvalues). It is interesting to consider how the entries denoted * in (1) can be normalized. In thefirst form, the eigenvalue 1 has geometric multiplicity mg equal to 1 or 2 (i.e., there are eitherone or two independent eigenvectors with eigenvalue 1). The normal form for mg = 2 hasbeen computed in [17]. The result is that only a single non-zero element remains above thediagonal. Essentially this means that when mg = 2, the matrix can be block-diagonalizedin SL(3, Z). In the remaining cases in (1) a block-diagonal form is in general not possible:conjugating a block triangular matrix with a block triangular matrix gives(1 −dD−10 D−1) (1 a0 A)(1 d0 D)=(1 (a − dD−1(A − 1))D0 D−1AD).Setting the upper right element of the right-hand side to zero and solving for d involves theinverse of A − 1 which is in general not an integer matrix. Using a more general transformation4 For general n the multiplicity cannot be n − 1.L446 Letter to the Editorleads to the same condition. Thus for general a and A, the monodromy matrix cannot be block-diagonalized. However, e.g., for the special matrix A = (2 11 1) (the ‘cat map’), it is alwayspossible since det(A − 1) = −1. If A − 1 is singular (corresponding to the first case incorollary 4), the resulting Diophantine equation may or may not have a solution.Our results need the condition µ = 0. It may be possible to show that µ = 0 necessarilyholds for certain configuration spaces M. We suspect, for example, that this is the case whenM = Rn, although we have not been able to prove this. Our result would then give thecomplete classification of monodromy matrices on T ∗Rn. In particular, the constructionof arbitrary monodromy matrices given in [6] would be impossible on these cotangentbundles.In 3 degrees of freedom, the known examples of monodromy are either of the first formwith mg = 2 [17, 18] or of the last form and block-diagonal. The last form of M is realizedfor geodesic flows on Sol-manifolds, where an arbitrary hyperbolic B ∈ SL(2, Z) mayappear [3].The main implication of the above is that when ma = 3 and mg = 2 there are always twoinvariant actions, i.e. actions that do not change globally along the path C. Obviously, thereis always one invariant action, namely the one corresponding to the eigenvector e1, and whenmg = 1 it is the only one. With eigenvalues −1 there is at most one invariant action, butanother action is invariant when C is traversed twice. Hence, on a covering space this mayreduce to ma = 3 and mg = 2. It would be very interesting to find an example of this type.AcknowledgmentsHRD would like to thank AP Veselov and VS Matveev for helpful discussions. This workwas supported by the EU Research Training Network ‘Mechanics and Symmetry in Europe’(MASIE) (HPRN CT-2000-00113).References[1] Arnol’d V I 1967 On a characteristic class entering into conditions of quantization Funct. Anal. Appl. 1 1–13[2] Arnol’d V I and Givental’ A B 2001 Symplectic geometry Dynamical Systems IV (Encyclopaedia Math. Sci.,vol 4) (Berlin: Springer) pp 1–138[3] Bolsinov A V, Dullin H R and Veselov A P 2005 Sol-manifolds: spectrum and quantum monodromyhttp://arXiv.org/abs/math-ph/0503046[4] Cassels J W S 1997 An introduction to the geometry of numbers Classics in Mathematics (Berlin: Springer)(Corrected reprint of the 1971 edition)[5] Child M S 1998 Quantum states in a champagne bottle J. Phys. A: Math. Gen. 31 657–70[6] Cushman R and Vũ Ngo. c S 2002 Sign of the monodromy for Liouville integrable systems Ann. Henri Poincaré3 883–94[7] Duistermaat J J 1980 On global action-angle coordinates Commun. Pure Appl. Math. 33 687–706[8] Dullin H R, Giacobbe A and Cushman R 2004 Monodromy in the resonant swing spring Physica D 190 15–37[9] Efstathiou K, Cushman R H and Sadovskiı́ D A 2004 Hamiltonian Hopf bifurcation of the hydrogen atom incrossed fields Physica D 194 250–74[10] Foxman J A and Robbins J M 2004 The Maslov index and nondegenerate singularities of integrable systemshttp://arxiv.org/abs/math-ph/0411018[11] Cushman R H and Duistermaat J J 1988 The quantum mechanical spherical pendulum Bull. Am. Math. Soc. 19475–9[12] Maslov V P and Fedoriuk M 1981 Semiclassical Approximation in Quantum Mechanics (Dordrecht: Reidel)[13] Matveev V S 1996 Integrable Hamiltonian systems with two degrees of freedom. Topological structure ofsaturated neighborhoods of saddle–saddle and focus–focus types Matem. Sbornik 187 29–58[14] Sadovskiı́ D A and Zĥilinskiı́ B I 1999 Monodromy, diabolic points, and angular momentum coupling Phys.Lett. A 256 235–44Letter to the Editor L447[15] Trofimov V V 1995 Generalized Maslov classes on the path space of a symplectic manifold Proc. Steklov. Inst.Math. 205 157–79[16] Vũ Ngo. c S 1999 Quantum monodromy in integrable systems Commun. Math. Phys. 203 465–79[17] Waalkens H and Dullin H R 2002 Quantum monodromy in prolate ellipsoidal billiards Ann. Phys. 295 81–112[18] Waalkens H, Dullin H R and Richter P H 2004 The problem of two fixed centers: bifurcations, actions,monodromy Physica D 196 265–310[19] Zung N T 1997 A note on focus–focus singularities Differ. Geom. Appl. 7 123–30

https://repository.lboro.ac.uk/articles/Maslov_indices_and_monodromy/9380216/files/16991744.pdf

Maslov Indices and Monodromy

Maslov Indices and Monodromy

Abstract

Similar works

Full text

Available Versions

Explore Bristol Research

University of Groningen Research Database

University of Groningen

Loughborough University Institutional Repository

NARCIS

Crossref

University of Groningen