Integrable Systems of Classical Mechanics and Lie Algebras [E-Book] : Volume I / by A. M. Perelomov.
Perelomov, A. M., (author)
Basel : Birkhäuser, 1990
online resource.
englisch
9783034892575
10.1007/978-3-0348-9257-5
Full Text
Table of Contents:
  • 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.