Integrable Systems of Classical Mechanics and Lie Algebras [E-Book] : Volume I / by A. M. Perelomov.
Saved in:
Full text |
|
Personal Name(s): | Perelomov, A. M., author |
Imprint: |
Basel :
Birkhäuser,
1990
|
Physical Description: |
online resource. |
Note: |
englisch |
ISBN: |
9783034892575 |
DOI: |
10.1007/978-3-0348-9257-5 |
Subject (LOC): |
- 1. Preliminaries
- 1.1 A Simple Example: Motion in a Potential Field
- 1.2 Poisson Structure and Hamiltonian Systems
- 1.3 Symplectic Manifolds
- 1.4 Homogeneous Symplectic Spaces
- 1.5 The Moment Map
- 1.6 Hamiltonian Systems with Symmetry
- 1.7 Reduction of Hamiltonian Systems with Symmetry
- 1.8 Integrable Hamiltonian Systems
- 1.9 The Projection Method
- 1.10 The Isospectral Deformation Method
- 1.11 Hamiltonian Systems on Coadjoint Orbits of Lie Groups
- 1.12 Constructions of Hamiltonian Systems with Large Families of Integrals of Motion
- 1.13 Completeness of Involutive Systems
- 1.14 Hamiltonian Systems and Algebraic Curves
- 2. Simplest Systems
- 2.1 Systems with One Degree of Freedom
- 2.2 Systems with Two Degrees of Freedom
- 2.3 Separation of Variables
- 2.4 Systems with Quadratic Integrals of Motion
- 2.5 Motion in a Central Field
- 2.6 Systems with Closed Trajectories
- 2.7 The Harmonic Oscillator
- 2.8 The Kepler Problem
- 2.9 Motion in Coupled Newtonian and Homogeneous Fields
- 2.10 Motion in the Field of Two Newtonian Centers
- 3. Many-Body Systems
- 3.1 Lax Representation for Many-Body Systems
- 3.2 Completely Integrable Many-Body Systems
- 3.3 Explicit Integration of the Equations of Motion for Systems of Type I and V via the Projection Method
- 3.4 Relationship Between the Solutions of the Equations of Motion for Systems of Type I and V
- 3.5 Explicit Integration of the Equations of Motion for Systems of Type II and III
- 3.6 Integration of the Equations of Motion for Systems with Two Types of Particles
- 3.7 Many-Body Systems as Reduced Systems
- 3.8 Generalizations of Many-Body Systems of Type I-III to the Case of the Root Systems of Simple Lie Algebras
- 3.9 Complete Integrability of the Systems of Section 3.8
- 3.10 Anisotropic Harmonic Oscillator in the Field of a Quartic Central Potential (the Garnier System)
- 3.11 A Family of Integrable Quartic Potentials Related to Symmetric Spaces
- 4. The Toda Lattice
- 4.1 The Ordinary Toda Lattice. Lax Representation. Complete Integrability
- 4.2 The Toda Lattice as a Dynamical System on a Coadjoint Orbit of the Group of Triangular Matrices
- 4.3 Explicit Integration of the Equations of Motion for the Ordinary Nonperiodic Toda Lattice
- 4.4 The Toda Lattice as a Reduced System
- 4.5 Generalized Nonperiodic Toda Lattices Related to Simple Lie Algebras
- 4.6 Toda-like Systems on Coadjoint Orbits of Borel Subgroups
- 4.7 Canonical Coordinates for Systems of Toda Type
- 4.8 Integrability of Toda-like Systems on Generic Orbits
- 5. Miscellanea
- 5.1 Equilibrium Configurations and Small Oscillations of Some Integrable Hamiltonian Systems
- 5.2 Motion of the Poles of Solutions of Nonlinear Evolution Equations and Related Many-Body Problems
- 5.3 Motion of the Zeros of Solutions of Linear Evolution Equations and Related Many-Body Problems
- 5.4 Concluding Remarks
- Appendix A
- Examples of Symplectic Non-Kählerian Manifolds
- Appendix B
- Solution of the Functional Equation (3.1.9)
- Appendix C
- Semisimple Lie Algebras and Root Systems
- Appendix D
- Symmetric Spaces
- References.