Table Of ContentINTEGRABLE HAMILTONIAN
SYSTEMS
Geometry, Topology, Classification
A. V. Bolsinov and A. T. Fomenko
CHAPMAN & HALL/CRC
A CRC Press Company
Boca Raton London New York Washington, D.C.
Copyright 2004 by CRC Press LL
INTEGRABLE HAMILTONIAN
SYSTEMS
Geometry, Topology, Classification
Copyright 2004 by CRC Press LL
TF1576 disclaimer.fm Page 1 Tuesday, January 20, 2004 3:53 PM
Library of Congress Cataloging-in-Publication Data
Bolsinov, A. V. (Aleksei Viktorovich)
[Integriruemye gamil’tonovy sistemy. English]
Integrable Hamiltonian systems : geometry, topology, classi(cid:222)cation / by A.V. Bolsinov,
A.T. Fomenko.
p. cm.
Includes bibliographical references and index.
ISBN 0-415-29805-9 (alk. paper)
1. Hamiltonian systems. 2. Geodesic (cid:223)ows. 3. Geodesics (Mathematics) I. Fomenko, A.
T. II. Title.
QA614.83.B6413 2004
515¢.39(cid:151)dc22 2003067457
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Contents
Preface
Chapter 1. Basic Notions
Section 1.1. Linear Symplectic Geometry
Section 1.2. SymplecticandPoisson Manifolds
1.2.1. Cotangent Bundles
n
1.2.2. The ComplexSpace C andIts ComplexSubmanifolds.
K(cid:127)ahler Manifolds
1.2.3. Orbits of Coadjoint Representation
Section 1.3. The Darboux Theorem
Section 1.4. Liouville Integrable HamiltonianSystems. The Liouville Theorem
Section 1.5. Non-resonantand ResonantSystems
Section 1.6. Rotation Number
Section 1.7. The MomentumMapping of an Integrable Systemand
Its Bifurcation Diagram
Section 1.8. Non-degenerate Critical Points of theMomentumMapping
1.8.1. The Case of Two Degrees of Freedom
1.8.2. Bott Integrals fromthe Viewpoint of theFour-Dimensional
Symplectic Manifold
1.8.3. Non-degenerate Singularities in the Case of Many Degrees
of Freedom
1.8.4. Types of Non-degenerate Singularities
in theMultidimensional Case
Section 1.9. Main Typesof Equivalence of DynamicalSystems
Chapter 2. The Topology of Foliations on Two-Dimensional Surfaces
Generated by Morse Functions
Section 2.1. SimpleMorse Functions
Section 2.2. ReebGraph of a Morse Function
Section 2.3. Notion of anAtom
Section 2.4. SimpleAtoms
2.4.1. The Case of MinimumandMaximum. The Atom A
2.4.2. The Case of anOrientable Saddle. The Atom B e
2.4.3. The Case of a Non-orientable Saddle. The Atom B
2.4.4. The Classi(cid:12)cation of SimpleAtoms
Copyright 2004 by CRC Press LL
Section 2.5. SimpleMolecules
2.5.1. Notion of a SimpleMolecule
2.5.2. Realization Theorem
2.5.3. Examples of SimpleMorse FunctionsandSimpleMolecules
2.5.4. The Classi(cid:12)cation of Minimal SimpleMorse Functions
on Surfaces of Low Genus
Section 2.6. Complicated Atoms
Section 2.7. Classi(cid:12)cation of Atoms
2.7.1. Classi(cid:12)cation Problem
2.7.2. Algorithm of Enumerationof Atoms
2.7.3. Algorithm of Recognition of Identical Atoms
2.7.4. Atomsand f-Graphs
2.7.5. Oriented AtomsandSubgroupsin theGroup Z(cid:3)Z2
2.7.6. Representation of Atomsas Immersionsof Graphs
into thePlane
2.7.7. Atomsas Cell Decompositions of Two-Dimensional Closed
Surfaces
2.7.8. Table of Atomsof Low Complexity
2.7.9. Mirror-like Atoms
Section 2.8. Symmetry Groups of Oriented Atoms and the Universal Covering
Tree
2.8.1. Symmetriesof f-Graphs
2.8.2. The Universal Covering Tree over f-Graphs. An f-Graphas
a Quotient Spaceof theUniversal Tree
2.8.3. The Correspondence between f-Graphsand Subgroups
in Z(cid:3)Z2
2.8.4. The Graph J of the SymmetryGroup of an f-Graph.
Totally Symmetric f-Graphs
2.8.5. The List of Totally SymmetricPlanar Atoms. Examplesof
Totally SymmetricAtomsof Genus g>0
2.8.6. Atomsas Surfaces of Constant Negative Curvature
Section 2.9. Notion of a Molecule
Section 2.10. Approximationof Complicated Molecules bySimpleOnes
Section 2.11. Classi(cid:12)cation ofMorse{SmaleFlowsonTwo-DimensionalSurfaces
byMeans of AtomsandMolecules
Chapter 3. Rough Liouville Equivalence of Integrable Systems
with Two Degrees of Freedom
Section 3.1. Classi(cid:12)cation of Non-degenerateCritical Submanifolds
on Isoenergy 3-Surfaces
Section 3.2. The Topological Structureof a Neighborhood of a Singular Leaf
Section 3.3. Topologically StableHamiltonian Systems
Section 3.4. Exampleof a Topologically Unstable Integrable System
Section 3.5. 2-Atomsand3-Atoms
Section 3.6. Classi(cid:12)cation of 3-Atoms
Section 3.7. 3-Atomsas Bifurcations of Liouville Tori
Section 3.8. The Molecule of anIntegrable System
Section 3.9. Complexityof Integrable Systems
Chapter 4. Liouville Equivalence of Integrable Systems
with Two Degrees of Freedom
Section 4.1. Admissible Coordinate Systemson theBoundaryof a 3-Atom
Section 4.2. Gluing Matrices andSuper(cid:13)uousFrames
Section 4.3. Invariants(Numerical Marks) r, ", and n
4.3.1. Marks ri and "i
4.3.2. Marks nk andFamilies in a Molecule
Copyright 2004 by CRC Press LL
Section 4.4. The Marked Molecule is a Complete Invariantof Liouville
Equivalence
Section 4.5. The In(cid:13)uenceof the Orientation
4.5.1. Change of Orientation of anEdge of a Molecule
4.5.2. Change of Orientation on a 3-Manifold Q
4.5.3. Change of Orientation of a HamiltonianVector Field
Section 4.6. Realization Theorem
Section 4.7. SimpleExamplesof Molecules
Section 4.8. Hamiltonian Systemswith Critical KleinBottles
Section 4.9. Topological Obstructionsto Integrability of Hamiltonian Systems
with Two Degrees of Freedom
4.9.1. The Class (M)
4.9.2. The Class (H)
4.9.3. The Class (Q)
4.9.4. The Class (W) of Graph-Manifolds
0
4.9.5. The Class (H) of Manifolds Related toHamiltonians
with Tame Integrals
4.9.6. The Coincidence of the FourClasses of 3-Manifolds
Chapter 5. Orbital Classi(cid:12)cation of Integrable Systems
with Two Degrees of Freedom
Section 5.1. Rotation Functionand Rotation Vector
Section 5.2. Reductionof theThree-Dimensional Orbital Classi(cid:12)cation
to theTwo-Dimensional Classi(cid:12)cation upto Conjugacy
5.2.1. Transversal Sections
5.2.2. Poincar(cid:19)e Flow andPoincar(cid:19)e Hamiltonian
5.2.3. ReductionTheorem
Section 5.3. General Concept of Constructing Orbital Invariants of Integrable
Hamiltonian Systems
Chapter 6. Classi(cid:12)cation of Hamiltonian Flows on Two-Dimensional
Surfaces up to Topological Conjugacy
Section 6.1. Invariantsof a HamiltonianSystemon a 2-Atom
6.1.1. (cid:3)-Invariant
6.1.2. (cid:1)-Invariantand Z-Invariant
Section 6.2. Classi(cid:12)cation of HamiltonianFlows withOne Degree of Freedom
upto Topological Conjugacy on Atoms
Section 6.3. Classi(cid:12)cation of HamiltonianFlows on2-Atomswith Involution
upto Topological Conjugacy
Section 6.4. The Pasting-Cutting Operation
Section 6.5. DescriptionoftheSetsofAdmissible(cid:1)-InvariantsandZ-Invariants
Chapter 7. Smooth Conjugacy of Hamiltonian Flows
on Two-Dimensional Surfaces
Section 7.1. Constructing SmoothInvariantson 2-Atoms
Section 7.2. Theorem of Classi(cid:12)cation of HamiltonianFlows on Atoms
upto SmoothConjugacy
Chapter 8. Orbital Classi(cid:12)cation of Integrable Hamiltonian Systems
with Two Degrees of Freedom. The Second Step
Section 8.1. Super(cid:13)uous t-Frameof a Molecule (Topological Case).
The Main Lemmaon t-Frames
Section 8.2. The Group of Transformations of Transversal Sections.
Pasting-Cutting Operation
Section 8.3. The Action of GP onthe Setof Super(cid:13)uous t-Frames
Copyright 2004 by CRC Press LL
Section 8.4. Three General Principles for Constructing Invariants
8.4.1. First General Principle
8.4.2. Second General Principle
8.4.3. Third General Principle
Section 8.5. Admissible Super(cid:13)uous t-Framesanda Realization Theorem
8.5.1. Realization of a Frame on an Atom
8.5.2. Realization of a Frame on an Edge of a Molecule
8.5.3. Realization of a Frame on theWhole Molecule
Section 8.6. Construction of Orbital Invariantsin theTopological Case.
A t-Molecule
8.6.1. The R-InvariantandtheIndexof a Systemon anEdge
8.6.2. be-Invariant(ontheRadicals of a Molecule)
8.6.3. (cid:3)e-Ienvaeriant
8.6.4. (cid:1)Z[(cid:2)]-Invariant
8.6.5. Final De(cid:12)nitionof a t-Molecule for anIntegrable System
Section 8.7. Theorem onthe Topological Orbital Classi(cid:12)cation of Integrable
Systemswith Two Degrees of Freedom
Section 8.8. AParticular Case: Simple Integrable Systems
Section 8.9. Smooth Orbital Classi(cid:12)cation
Chapter 9. Liouville Classi(cid:12)cation of Integrable Systems with
Two Degrees of Freedom in Four-Dimensional
Neighborhoods of Singular Points
Section 9.1. l-Type of a Four-DimensionalSingularity
Section 9.2. The Loop Molecule of a Four-DimensionalSingularity
Section 9.3. Center{Center Case
Section 9.4. Center{Saddle Case
Section 9.5. Saddle{Saddle Case
9.5.1. The Structure of a Singular Leaf
9.5.2. Cl-Typeof a Singularity
9.5.3. TheListofSaddle{SaddleSingularitiesofSmallComplexity
Section 9.6. AlmostDirect Product Representation of a Four-Dimensional
Singularity
Section 9.7. Proof of theClassi(cid:12)cation Theorems
9.7.1. Proof of Theorem 9.3
9.7.2. Proof of Theorem 9.4 (Realization Theorem)
Section 9.8. Focus{Focus Case
9.8.1. The Structureof a Singular Leaf of Focus{Focus Type
9.8.2. Classi(cid:12)cation of Focus{Focus Singularities
9.8.3. Model Exampleof a Focus{Focus Singularity and
the Realization Theorem
9.8.4. The Loop Molecule andMonodromyGroupof a Focus{Focus
Singularity
Section 9.9. AlmostDirect Product Representation for Multidimensional
Non-degenerate Singularities of Liouville Foliations
Chapter 10. Methods of Calculation of Topological Invariants
of Integrable Hamiltonian Systems
Section 10.1. GeneralSchemeforTopologicalAnalysisoftheLiouvilleFoliation
10.1.1. MomentumMapping
10.1.2. Construction of the Bifurcation Diagram
10.1.3. Veri(cid:12)cation of theNon-degeneracyCondition
10.1.4. Description of theAtomsof the System
10.1.5. Construction of the Molecule of the Systemona Given
Energy Level
10.1.6. Computation of Marks
Copyright 2004 by CRC Press LL
Section 10.2. Methods for ComputingMarks
Section 10.3. The Loop Molecule Method
Section 10.4. List of Typical Loop Molecules
10.4.1. Loop Molecules of Regular Points of the Bifurcation
Diagram
10.4.2. Loop Molecules of Non-degenerateSingularities
Section 10.5. The Structureof theLiouville Foliation for Typical Degenerate
Singularities
Section 10.6. Typical Loop Molecules Corresponding to Degenerate
One-DimensionalOrbits
Section 10.7. Computationof r-and "-MarksbyMeansofRotationFunctions
Section 10.8. Computationof the n-Mark byMeans of Rotation Functions
Section 10.9. Relationship Between the Marks of theMolecule and
3
the Topology of Q
Chapter 11. Integrable Geodesic Flows on Two-Dimensional Surfaces
Section 11.1. Statementof the Problem
Section 11.2. Topological Obstructions to Integrability of Geodesic Flows
onTwo-Dimensional Surfaces
Section 11.3. Two Examplesof Integrable Geodesic Flows
11.3.1. Surfaces of Revolution
11.3.2. Liouville Metrics
Section 11.4. Riemannian Metrics Whose Geodesic Flows are Integrable
byMeans of Linear or Quadratic Integrals. Local Theory
11.4.1. SomeGeneral Properties of Polynomial Integrals
of Geodesic Flows. Local Theory
11.4.2. RiemannianMetricsWhoseGeodesicFlowsAdmitaLinear
Integral. Local Theory
11.4.3. RiemannianMetrics Whose Geodesic Flows Admit
a Quadratic Integral. Local Theory
Section 11.5. Linearly andQuadratically IntegrableGeodesic Flows onClosed
Surfaces
11.5.1. The Torus
11.5.2. The Klein Bottle
11.5.3. The Sphere
11.5.4. The Projective Plane
Chapter 12. Liouville Classi(cid:12)cation of Integrable Geodesic Flows
on Two-Dimensional Surfaces
Section 12.1. The Torus
Section 12.2. The KleinBottle
12.2.1. Quadratically Integrable Geodesic Flow onthe Klein
Bottle
12.2.2. Linearly IntegrableGeodesic Flows onthe KleinBottle
12.2.3. Quasi-Linearly IntegrableGeodesic Flows onthe Klein
Bottle
12.2.4. Quasi-QuadraticallyIntegrableGeodesicFlowsontheKlein
Bottle
Section 12.3. The Sphere
12.3.1. Quadratically Integrable Geodesic Flows on the Sphere
12.3.2. Linearly IntegrableGeodesic Flows onthe Sphere
Section 12.4. The Projective Plane
12.4.1. Quadratically Integrable Geodesic Flows on the Projective
Plane
12.4.2. Linearly Integrable Geodesic Flows onthe Projective
Plane
Copyright 2004 by CRC Press LL
Chapter 13. Orbital Classi(cid:12)cation of Integrable Geodesic Flows
on Two-Dimensional Surfaces
Section 13.1. Case of theTorus
13.1.1. Flows withSimple Bifurcations (Atoms)
13.1.2. Flows withComplicated Bifurcations (Atoms)
Section 13.2. Case of theSphere
Section 13.3. Examplesof Integrable Geodesic Flows on the Sphere
13.3.1. The Triaxial Ellipsoid
13.3.2. The StandardSphere
13.3.3. The Poisson Sphere
Section 13.4. Non-trivialityof OrbitalEquivalence Classes andMetrics with
Closed Geodesics
Chapter 14. TheTopologyofLiouvilleFoliationsinClassicalIntegrable
Cases in Rigid Body Dynamics
Section 14.1. Integrable Cases inRigid Body Dynamics
Section 14.2. Topological Type of Isoenergy 3-Surfaces
14.2.1. The Topology ofthe IsoenergySurfaceandtheBifurcation
Diagram
14.2.2. Euler Case
14.2.3. Lagrange Case
14.2.4. KovalevskayaCase
14.2.5. Zhukovski(cid:20)(cid:16) Case
14.2.6. Goryachev{Chaplygin{Sretenski(cid:20)(cid:16) Case
14.2.7. Clebsch Case
14.2.8. SteklovCase
Section 14.3. Liouville Classi(cid:12)cation of Systemsin theEuler Case
Section 14.4. Liouville Classi(cid:12)cation of Systemsin the Lagrange Case
Section 14.5. Liouville Classi(cid:12)cation of Systemsin theKovalevskayaCase
Section 14.6. Liouville Classi(cid:12)cation of Systems
inthe Goryachev{Chaplygin{Sretenski(cid:20)(cid:16) Case
Section 14.7. Liouville Classi(cid:12)cation of Systemsin the Zhukovski(cid:20)(cid:16) Case
Section 14.8. RoughLiouville Classi(cid:12)cation of Systemsinthe Clebsch Case
Section 14.9. RoughLiouville Classi(cid:12)cation of Systemsinthe SteklovCase
Section 14.10. RoughLiouville Classi(cid:12)cation of Integrable Four-Dimensional
Rigid Body Systems
Section 14.11. The Complete List of Molecules Appearing in Integrable Cases
of Rigid Body Dynamics
Chapter 15. Maupertuis Principle and Geodesic Equivalence
Section 15.1. General Maupertuis Principle
Section 15.2. Maupertuis Principle in Rigid Body Dynamics
Section 15.3. Classical Cases of Integrability in Rigid Body Dynamicsand
Related IntegrableGeodesic Flows onthe Sphere
15.3.1. Euler Case and thePoisson Sphere
15.3.2. Lagrange Case and Metrics of Revolution
15.3.3. Clebsch Case andGeodesic Flow onthe Ellipsoid
15.3.4. Goryachev{Chaplygin Case andthe Corresponding
Integrable Geodesic Flow on theSphere
15.3.5. KovalevskayaCase and the Corresponding Integrable
Geodesic Flow on theSphere
Section 15.4. Conjecture on Geodesic Flows with Integrals of High Degree
Section 15.5. DiniTheorem andthe Geodesic Equivalence of Riemannian
Metrics
Section 15.6. Generalized Dini{Maupertuis Principle
Copyright 2004 by CRC Press LL
Section 15.7. Orbital Equivalence of theNeumannProblemandthe Jacobi
Problem
Section 15.8. Explicit Formsof SomeRemarkableHamiltonians andTheir
Integrals in Separating Variables
Chapter 16. Euler Case in Rigid Body Dynamics and Jacobi Problem
about Geodesics on the Ellipsoid. Orbital Isomorphism
Section 16.1. Introduction
Section 16.2. Jacobi Problemand Euler Case
Section 16.3. Liouville Foliations
Section 16.4. Rotation Functions
Section 16.5. The Main Theorem
Section 16.6. SmoothInvariants
Section 16.7. Topological Non-conjugacy of the Jacobi Problem andthe Euler
Case
References
Copyright 2004 by CRC Press LL
