This book illustrates the powerful interplay between topological, algebraic and complex analytical methods, within the field of integrable systems, by addressing several theoretical and practical aspects. Contemporary integrability results, discovered in the last few decades, are used within different areas of mathematics and physics.
Integrable Systems incorporates numerous concrete examples and exercises, and covers a wealth of essential material, using a concise yet instructive approach. This book is intended for a broad audience, ranging from mathematicians and physicists to students pursuing graduate, Masters or further degrees in mathematics and mathematical physics. It also serves as an excellent guide to more advanced and detailed reading in this fundamental area of both classical and contemporary mathematics.
Author(s): Ahmed Lesfari
Series: Mathematics and Statistics
Edition: 1
Publisher: Wiley-ISTE
Year: 2022
Language: English
Pages: 308
Tags: Integrable Systems, Symplectic Manifolds, Hamilton-Jacobi Theory, Spectral Methods, Flows, KdV-Equation, Pseudo-differential Operators
Cover
Half-Title Page
Dedication
Title Page
Copyright Page
Contents
Preface
Chapter 1. Symplectic Manifolds
1.1. Introduction
1.2. Symplectic vector spaces
1.3. Symplectic manifolds
1.4. Vectors fields and flows
1.5. The Darboux theorem
1.6. Poisson brackets and Hamiltonian systems
1.7. Examples
1.8. Coadjoint orbits and their symplectic structures
1.9. Application to the group SO(n)
1.9.1. Application to the group SO(3)
1.9.2. Application to the group SO(4)
1.10. Exercises
Chapter 2. Hamilton–Jacobi Theory
2.1. Euler–Lagrange equation
2.2. Legendre transformation
2.3. Hamilton’s canonical equations
2.4. Canonical transformations
2.5. Hamilton–Jacobi equation
2.6. Applications
2.6.1. Harmonic oscillator
2.6.2. The Kepler problem
2.6.3. Simple pendulum
2.7. Exercises
Chapter 3. Integrable Systems
3.1. Hamiltonian systems and Arnold–Liouville theorem
3.2. Rotation of a rigid body about a fixed point
3.2.1. The Euler problem of a rigid body
3.2.2. The Lagrange top
3.2.3. The Kowalewski spinning top
3.2.4. Special cases
3.3. Motion of a solid through ideal fluid
3.3.1. Clebsch’s case
3.3.2. Lyapunov–Steklov’s case
3.4. Yang–Mills field with gauge group SU(2)
3.5. Appendix (geodesic flow and Euler–Arnold equations)
3.6. Exercises
Chapter 4. Spectral Methods for Solving Integrable Systems
4.1. Lax equations and spectral curves
4.2. Integrable systems and Kac–Moody Lie algebras
4.3. Geodesic flow on SO(n)
4.4. The Euler problem of a rigid body
4.5. The Manakov geodesic flow on the group SO(4)
4.6. Jacobi geodesic flow on an ellipsoid and Neumann problem
4.7. The Lagrange top
4.8. Quartic potential, Garnier system
4.9. The coupled nonlinear Schrödinger equations
4.10. The Yang–Mills equations
4.11. The Kowalewski top
4.12. The Goryachev–Chaplygin top
4.13. Periodic infinite band matrix
4.14. Exercises
Chapter 5. The Spectrum of Jacobi Matrices and Algebraic Curves
5.1. Jacobi matrices and algebraic curves
5.2. Difference operators
5.3. Continued fraction, orthogonal polynomials and Abelian integrals
5.4. Exercises
Chapter 6. Griffiths Linearization Flows on Jacobians
6.1. Spectral curves
6.2. Cohomological deformation theory
6.3. Mittag–Leffler problem
6.4. Linearizing flows
6.5. The Toda lattice
6.6. The Lagrange top
6.7. Nahm’s equations
6.8. The n-dimensional rigid body
6.9. Exercises
Chapter 7. Algebraically Integrable Systems
7.1. Meromorphic solutions
7.2. Algebraic complete integrability
7.3. The Liouville–Arnold–Adler–van Moerbeke theorem
7.4. The Euler problem of a rigid body
7.5. The Kowalewski top
7.6. The Hénon–Heiles system
7.7. The Manakov geodesic flow on the group SO(4)
7.8. Geodesic flow on SO(4) with a quartic invariant
7.9. The geodesic flow on SO(n) for a left invariant metric
7.10. The periodic five-particle Kac–van Moerbeke lattice
7.11. Generalized periodic Toda systems
7.12. The Gross–Neveu system
7.13. The Kolossof potential
7.14. Exercises
Chapter 8. Generalized Algebraic Completely Integrable Systems
8.1. Generalities
8.2. The RDG potential and a five-dimensional system
8.3. The Hénon–Heiles problem and a five-dimensional system
8.4. The Goryachev–Chaplygin top and a seven-dimensional system
8.5. The Lagrange top
8.6. Exercises
Chapter 9. The Korteweg–de Vries Equation
9.1. Historical aspects and introduction
9.2. Stationary Schrödinger and integral Gelfand–Levitan equations
9.3. The inverse scattering method
9.4. Exercises
Chapter 10. KP–KdV Hierarchy and Pseudo-differential Operators
10.1. Pseudo-differential operators and symplectic structures
10.2. KdV equation, Heisenberg and Virasoro algebras
10.3. KP hierarchy and vertex operators
10.4. Exercises
References
Index
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