"Integrable Systems on Lie Algebras and Symmetric Spaces" by A. T. Fomenko, V. V. Trofimov
Advanced Studies in Contemporary Mathematics, volume 2

Gordon and Breach Science Publishers | 1988 | ISBN: 2881241700 9782881241703 | 307 pages | PDF | 3 MB
This book sets out some new methods for integrating Hamilton's canonical equations. Common to all these methods is one overall idea: the realization of canonical equations in Lie algebras or symmetric spaces.

Basically, the book sets out results obtained by the authors and by participants in the scientific research seminar Contemporary Geometry Methods, run at Moscow University under the direction of A.T. Fomenko.

Contents
Introduction
1. Symplectic Geometry and the Integration of Hamiltonian Systems
1. Symplectic manifolds
1.1. Symplectic Structure and its Canonical Representation. Skew-Symmetric Gradient
1.2. The Geometric Properties of Symplectic Structures
1.3. Hamiltonian Vector Fields
1.4. The Poisson Bracket and Hamiltonian Field Integrals
1.5. Degenerate Poisson Brackets
2. Symplectic Geometries and Lie Groups
2.1. Summary of the Necessary Results on Lie Groups and Lie Algebras
2.2. Orbits of the Coadjoint Representation and the Canonical Symplectic Structure
2.3. Differential Equations for Invariants and Semi-Invariants of the Coadjoint Representation
3. Liouville's Theorem
3.1. Commutative Integration of Hamiltonian Systems
3.2. Non-Commutative Lie Algebras of Integrals
3.3. Theorem of Non-Commutative Integration
3.4. Reduction of Hamiltonian Systems with Non-Commutative Symmetries
3.5. Orbits of the Coadjoint Representation as Symplectic Manifolds
3.6. The Connection between Commutative and Non-Commutative Liouville Integration
4. Algebraicization of Hami Ionian Systems on Lie Group Orbits
4.1. The Realization of Hamiltonian Systems on the Orbits of the Coadjoint Representation
4.2. Examples of Algebraicized Systems
5. Complete Commutative Sets of Functions on Symplectic Manifolds
2. Sectional Operators and Their Applications
6. Sectional Operators, Finite-Dimensional Representations, Dynamic Systems on the Orbits of Representation
7. Examples of Sectional Operators
7.1. Equations of Motion of a Multi-Dimensional Rigid Body with a Fixed Point and Their Analogs on Semi-Simple Lie Algebras. The Complex Semi-Simple Series
7.2. Hamiltonian Systems of the Compact and the Normal Series
7.3. Equations of Inertial Motion of a Multi-Dimensional Rigid Body in an Ideal Fluid
7.4. Equations of Inertial Motion of a Multi-Dimensional Rigid Body in an Incompressible Ideally Conductive Fluid
3. Sectional Operators on Symmetric Spaces
8. Construction of the Form Fc and the Flow XQ in the Case of a Symmetric Space
9. The Case of the Group S (Symmetric Spaces of Type II)
10. The Case of Type 1, III, IV Symmetric Spaces
10.1 Symmetric Spaces of Maximal Rank
10.2. The Symmetric Space Sn' = SO(n)/SO(n - 1) (The Real Case)
10.3. Hamiltonian Flows XQ, Symplectic Structures Fc and the Equations of Motion of Analogs of a Multi-Dimensional Rigid Body
10.4. The Symmetric Space S"-' = SO(n)/SO(n - 1) (The Complex Case)
10.5. Examples of Flows X, on S"-' (The Complex Case)
4. Methods of Construction of Functions in Involution on Orbits of Coadjoint Representation of Lie Groups
11. Method of Argument Translation
11.1. Translations of Invariants of Coadjoint Representation
11.2. Representations of Lie Groups in the Space of the Functions on the Orbits and Corresponding Involutive Sets of Functions
12. Methods of Construction of Commutative Sets of Functions Using Chains of Subalgebras
13. Method of Tensor Extensions of Lie Algebras
13.1. Basic Definitions and Results
13.2. The Proof of the General Theorem
13.3. The Application of the Algorithm (21) to the Construction of S-Representations
13.4. Algebras with Poincare Duality
14. Similar Functions
14.1. Partial Invariants
14.2. Involutivity of Similar Functions
15. Contractions of Lie Algebras
15.1 Restriction Theorem
15.2. Contractions of 7L2 -Graded Lie Algebras
5. Complete Integrability of Hamiltonian Systems on Orbits of Lie Algebras
16. Complete Integrability of the Equations of Motion of a Multi-Dimensional Rigid Body with a Fixed Point in the Absence of Gravity
16.1. Integrals of Euler Equations on Semi-Simple Lie Algebras
16.2. Examples for Lie Algebras of so(3) and so(4)
16.3. Cases of Complete Integrability of Euler's Equations on Semi-Simple Lie Algebras
17. Cases of Complete Integrability of the Equations of Inertial Motion of a Mufti-Dimensional Rigid Body in an Ideal Fluid
18. The Case of Complete Integrability of the Equations of Inertial Motion of a Multi-Dimensional Rigid Body in an Incompressible, Ideally Conductive Fluid
18.1. Complete Integrability of the Euler Equations on Extensions f2(G) of Semi-Simple Lie Algebras
18.3. Extensions of fl(G) for Low-Dimensional Lie Algebras
19. Some Integrable Hamiltonian Flows with Semi-Simple Group of Symmetries
19.1. Integrable Systems in the 'Compact Case'
19.2. Integrable Systems in the Non-Compact Case. Multi-Dimensional Lagrange's Case
19.3. Functional Independence of Integrals 212
20. The Integrability of Certain Hamiltonian Systems on Lie Algebras
20.1. Completely Involutive Sets of Functions on Singular Orbits in su(m)
20.2. Completely Involutive Sets of Functions on Affine Lie Algebras
21. Completely Involutive Sets of Functions on Extensions of Abelian Lie Algebras
21.1. The Main Construction
21.2. Lie. Algebras of Triangular Matrices
22. Integrability of Eider's Equations on Singular Orbits of Semi-Simple Lie Algebras
22.1. Integrability of Euler's Equations on Orbits 0 Intersecting the Set tHa, teC
22.2. Integrability of Euler's Equations x = [x, (J1abD(x)] for Singular a
22.3. Integrability of Euler's Equations z = [X,(QabD(x)] on the Subalgebra G. Fixed Under the Canonical Involutive Automorphism a:G-+G for Singular Elements aeG
22.4. Integrability of Euler's Equations for an n-Dimensional Rigid Body
23. Completely Integrable Hamiltonian Systems on Symmetric Spaces
23.1. Integrable Metrics dspbD on Symmetric Spaces
23.2. The Metrics dsob on a Sphere S"
23.3. Applications to Non-Commutative Integrability 263
24. Morse's Theory of Completely Integrable Hamiltonian Systems. Topology of the Surfaces of Constant Energy Level of Hamiltonian Systems, Obstacles to Integrability and Classification of the Rearrangements of the General Position of Liouville Tori in the Neighborhood of a Bifurcation Diagram
24.1. The Four-Dimensional Case
24.2. The General Case
Bibliography
Index