INTEGRABLE 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 This book contains information obtained from authentic and highly regarded sources. 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Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identi(cid:222)cation and explanation, without intent to infringe. Visit the CRC Press Web site at www.crcpress.com ' 2004 by Chapman & Hall/CRC No claim to original U.S. Government works International Standard Book Number 0-415-29805-9 Library of Congress Card Number 2003067457 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper Copyright 2004 by CRC Press LL 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