This book is designed to expose from a general and universal standpoint a variety ofmethods and results concerning integrable systems ofclassical me- chanics. By such systems we mean Hamiltonian systems with a finite number of degrees of freedom possessing sufficiently many conserved quantities (in- tegrals ofmotion) so that in principle integration ofthe correspondingequa- tions of motion can be reduced to quadratures, i.e. to evaluating integrals of known functions. The investigation of these systems was an important line ofstudy in the last century which, among other things, stimulated the appearance of the theory ofLie groups. Early in our century, however, the work ofH. Poincare made it clear that global integrals of motion for Hamiltonian systems exist only in exceptional cases, and the interest in integrable systems declined. Until recently, only a small number ofsuch systems with two or more de- grees of freedom were known.
In the last fifteen years, however, remarkable progress has been made in this direction due to the invention by Gardner, Greene, Kruskal, and Miura [GGKM 19671 ofa new approach to the integra- tion ofnonlinear evolution equations known as the inverse scattering method or the method of isospectral deformations. Applied to problems of mechanics this method revealed the complete in- tegrability of numerous classical systems. It should be pointed out that all systems of this kind discovered so far are related to Lie algebras, although often this relationship is not sosimpleas the oneexpressed by the well-known theorem of E. Noether.
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-Kahlerian Manifolds.- Appendix B.- Solution of the Functional Equation (3.1.9).- Appendix C.- Semisimple Lie Algebras and Root Systems.- Appendix D.- Symmetric Spaces.- References.