Integrable Systems

Table of contents

1. Cover
2. Dedication
3. Title Page
4. Copyright
5. Preface
6. 1 Symplectic Manifolds
   1. 1.1. Introduction
   2. 1.2. Symplectic vector spaces
   3. 1.3. Symplectic manifolds
   4. 1.4. Vectors fields and flows
   5. 1.5. The Darboux theorem
   6. 1.6. Poisson brackets and Hamiltonian systems
   7. 1.7. Examples
   8. 1.8. Coadjoint orbits and their symplectic structures
   9. 1.9. Application to the group SO(n)
   10. 1.10. Exercises
7. 2 Hamilton–Jacobi Theory
   1. 2.1. Euler–Lagrange equation
   2. 2.2. Legendre transformation
   3. 2.3. Hamilton’s canonical equations
   4. 2.4. Canonical transformations
   5. 2.5. Hamilton–Jacobi equation
   6. 2.6. Applications
   7. 2.7. Exercises
8. 3 Integrable Systems
   1. 3.1. Hamiltonian systems and Arnold–Liouville theorem
   2. 3.2. Rotation of a rigid body about a fixed point
   3. 3.3. Motion of a solid through ideal fluid
   4. 3.4. Yang–Mills field with gauge group SU(2)
   5. 3.5. Appendix (geodesic flow and Euler–Arnold equations)
   6. 3.6. Exercises
9. 4 Spectral Methods for Solving Integrable Systems
   1. 4.1. Lax equations and spectral curves
   2. 4.2. Integrable systems and Kac–Moody Lie algebras
   3. 4.3. Geodesic flow on SO(n)
   4. 4.4. The Euler problem of a rigid body
   5. 4.5. The Manakov geodesic flow on the group SO(4)
   6. 4.6. Jacobi geodesic flow on an ellipsoid and Neumann problem
   7. 4.7. The Lagrange top
   8. 4.8. Quartic potential, Garnier system
   9. 4.9. The coupled nonlinear Schrödinger equations
   10. 4.10. The Yang–Mills equations
   11. 4.11. The Kowalewski top
   12. 4.12. The Goryachev–Chaplygin top
   13. 4.13. Periodic infinite band matrix
   14. 4.14. Exercises
10. 5 The Spectrum of Jacobi Matrices and Algebraic Curves
    1. 5.1. Jacobi matrices and algebraic curves
    2. 5.2. Difference operators
    3. 5.3. Continued fraction, orthogonal polynomials and Abelian integrals
    4. 5.4. Exercises
11. 6 Griffiths Linearization Flows on Jacobians
    1. 6.1. Spectral curves
    2. 6.2. Cohomological deformation theory
    3. 6.3. Mittag–Leffler problem
    4. 6.4. Linearizing flows
    5. 6.5. The Toda lattice
    6. 6.6. The Lagrange top
    7. 6.7. Nahm’s equations
    8. 6.8. The n-dimensional rigid body
    9. 6.9. Exercises
12. 7 Algebraically Integrable Systems
    1. 7.1. Meromorphic solutions
    2. 7.2. Algebraic complete integrability
    3. 7.3. The Liouville–Arnold–Adler–van Moerbeke theorem
    4. 7.4. The Euler problem of a rigid body
    5. 7.5. The Kowalewski top
    6. 7.6. The Hénon–Heiles system
    7. 7.7. The Manakov geodesic flow on the group SO(4)
    8. 7.8. Geodesic flow on SO(4) with a quartic invariant
    9. 7.9. The geodesic flow on SO(n) for a left invariant metric
    10. 7.10. The periodic five-particle Kac–van Moerbeke lattice
    11. 7.11. Generalized periodic Toda systems
    12. 7.12. The Gross–Neveu system
    13. 7.13. The Kolossof potential
    14. 7.14. Exercises
13. 8 Generalized Algebraic Completely Integrable Systems
    1. 8.1. Generalities
    2. 8.2. The RDG potential and a five-dimensional system
    3. 8.3. The Hénon–Heiles problem and a five-dimensional system
    4. 8.4. The Goryachev–Chaplygin top and a seven-dimensional system
    5. 8.5. The Lagrange top
    6. 8.6. Exercises
14. 9 The Korteweg–de Vries Equation
    1. 9.1. Historical aspects and introduction
    2. 9.2. Stationary Schrödinger and integral Gelfand–Levitan equations
    3. 9.3. The inverse scattering method
    4. 9.4. Exercises
15. 10 KP–KdV Hierarchy and Pseudo-differential Operators
    1. 10.1. Pseudo-differential operators and symplectic structures
    2. 10.2. KdV equation, Heisenberg and Virasoro algebras
    3. 10.3. KP hierarchy and vertex operators
    4. 10.4. Exercises
16. References
17. Index
18. End User License Agreement