Integrable systems on Lie algebras and symmetric spaces

Contents......Page 5
Introduction......Page 11
1.1. Symplectic Structure and its Canonical Representation Skew-Symmetric Gradient......Page 13
1.2. The Geometric Properties of Symplectic Structures......Page 16
1.3. Hamiltonian Vector Fields......Page 20
1.4. The Poisson Bracket and Hamiltonian Field Integrals......Page 23
1.5. Degenerate Poisson Brackets......Page 27
2.1. Summary of the Necessary Results on Lie Groups and Lie Algebras......Page 29
2.2. Orbits of the Coadjoint Representation and the Canonical Symplectic Structure......Page 34
2.3. Differential Equations for Invariants and Semi-Invariants of the Coadjoint Representation......Page 39
3.1. Commutative Integration of Hamiltonian Systems......Page 42
3.2. Non-Commutative Lie Algebras of Integrals......Page 44
3.3. Theorem of Non-Commutative Integration......Page 46
3.4. Reduction of Hamiltonian Systems with Non-Commutative Symmetries......Page 48
3.5. Orbits of the Coadjoint Representation as Symplectic Manifolds......Page 58
3.6. The Connection between Commutative and Non-Commutative Liouville Integration......Page 59
4.1. The Realization of Hamiltonian Systems on the Orbits of the Coadjoint Representation......Page 64
4.2. Examples of Algebraicized Systems......Page 70
5. Complete Commutative Sets of Functions on Symplectic Manifolds......Page 75
6. Sectional Operators, Finite-Dimensional Representations, Dynamic Systems on the Orbits of Representation......Page 79
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......Page 83
7.2. Hamiltonian Systems of the Compact and the Normal Series......Page 87
7.3. Equations of Inertial Motion of a Multi-Dimensional Rigid Body in an Ideal Fluid......Page 91
7.4. Equations of Inertial Motion of a Multi-Dimensional Rigid Body in an Incompressible Ideally Conductive Fluid......Page 101
8. Construction of the Form Fc and the Flow XQ in the Case of a Symmetric Space......Page 112
9. The Case of the Group $3 = S = ($3 x 3)/$3 (Symmetric Spaces of Type II)......Page 117
10.1 Symmetric Spaces of Maximal Rank......Page 119
10.2. The Symmetric Space S\"-\' = SO(n)/SO(n - 1) (The Real Case)......Page 123
10.3. Hamiltonian Flows XQ, Symplectic Structures Fc and the Equations of Motion of Analogs of a Multi-Dimensional Rigid Body......Page 132
10.4. The Symmetric Space S\" -\' = SO(n)/SO(n - 1) (The Complex Case)......Page 133
10.5. Examples of Flows X Q on S\" -\' (The Complex Case)......Page 143
11.1. Translations of Invariants of Coadjoint Representation......Page 148
11.2. Representations of Lie Groups in the Space of the Functions on the Orbits and Corresponding Involutive Sets of Functions......Page 150
12. Methods of Construction of Commutative Sets of Functions Using Chains of Subalgebras......Page 155
13.1. Basic Definitions and Results......Page 159
13.2. The Proof of the General Theorem......Page 163
13.3. The Application of the Algorithm (111) to the Construction of S-Representations......Page 172
13.4. Algebras with Poincare Duality......Page 174
14.1. Partial Invariants......Page 179
14.2. involutivity of Similar Functions......Page 180
15.1 Restriction Theorem......Page 183
15.2. Contractions of 12 -Graded Lie Algebras......Page 186
16.1. Integrals of Euler Equations on Semi-Simple Lie Algebras......Page 191
16.2. Examples for Lie Algebras of so(3) and so(4)......Page 197
16.3. Cases of Complete Integrability of Euler\'s Equations on Semi-Simple Lie Algebras......Page 201
17. Cases of Complete Integrability of the Equations of Inertial Motion of a Multi-Dimensional Rigid Body in an Ideal Fluid......Page 206
18.1. Complete Integrability of the Euler Equations on Extensions ((G) of Semi-Simple Lie Algebras......Page 210
18.2. Complete Integrability of a Geodesic Flow on T*Q(%)......Page 215
18.3. Extensions of Q (G) for Low-Dimensional Lie Algebras......Page 216
19.1. Integrable Systems in the \'Compact Case\'......Page 217
19.2. Integrable Systems in the Non-Compact Case. MultiDimensional Lagrange\'s Case......Page 220
19.3. Functional Independence of Integrals......Page 224
20. The Integrability of Certain Hamiltonian Systems on Lie Algebras......Page 226
20.1. Completely Involutive Sets of Functions on Singular Orbits in su(m)......Page 227
20.2. Completely Involutive Sets of Functions on Affine Lie Algebras......Page 231
21.1. The Main Construction......Page 236
21.2. Lie Algebras of Triangular Matrices......Page 244
22.1. Integrability of Euler\'s Equations on Orbits 0 Intersecting the Set tH,,,, teC......Page 248
22.2. Integrability of Euler\'s Equations [x, for Singular a......Page 256
22.3. Integrability of Euler\'s Equations z = on the Subalgebra G\" Fixed Under the Canonical Involutive Automorphism o:G-\'G for Singular Elements aeG......Page 259
22.4. Integrability of Euler\'s Equations for an n-Dimensional Rigid Body......Page 265
23.1. Integrable Metrics dsubv on Symmetric Spaces......Page 266
23.2. The Metrics dsab on a Sphere S\"......Page 269
23.3. Applications to Non-Commutative Integrability......Page 275
24.1. The Four-Dimensional Case......Page 278
24.2. The General Case......Page 282
Bibliography......Page 293
Index......Page 305