PHYSICS REPORTS (Review Section of Physics Letters) 71, No. 5(1981) 313-400. North-Holland Publishing Company CLASSICAL INTEGRABLE FINiTE-DIMENSIONAL SYSTEMS RELATED TO LIE ALGEBRAS MA. OLSHANETSKY and A.M. PERELOMOV Institute of Theoret~caI and Experimental Physics, 117259 Moscow, USSR Received 23 October 1980 Contents: 0. Introduction 315 13. Explicit fonnulae and moment map for the abstract Hamil- 1. General description 319 tonian systems 359 2. Completely integrable Hamiltonian systems 320 14. Explicit integration of the equations of motion for the 3. Systems with additional integrals of motion 321 systems of type IV and VI’ (periodic Toda lattice) 364 4. Proof of complete integrability of the systems of section 3 324 14.1. The systems of type VI’ AN_i 364 5. Explicit integration of theequations of motion for potentials 14.2. The systems of type IV AN_I 368 V(q) of type I and V 327 15. Miscellanea 371 6. Explicit integration of theequations of motion for potentials 15.1. Motion of the poles of nonlinear partial differential of type II and III 331 equations and related many-body problems 371 7. Integration ofthe equations ofmotion for a system with two 15.2. Motion of the zeros of the linear evolution equa- types of particles 333 tions and related integrable many-body problems 374 8. Explicit integration of theequations of motion for the Toda 15.3. Rotation of a many-dimensional rigid body around a lattice 335 fixed point 376 9. Reduction of Hamiltonian systems with symmetries 15.4. Concluding remarks 380 (Methods of orbits) 337 Appendix A. Solution to the functional equation (3.9) 382 10. Equilibrium configurations and small oscillations of some Appendix B. Groups generated by reflections and root sys- dynamical systems 346 tems 385 11. Abstract Hamiltonian systems related to root systems 352 Appendix C. Symmetric spaces 390 12. Complete integrability of the abstract Hamiltonian systems 355 References 398 Abstract: During the last few years many dynam*cal systems have been identified, that are completely integrable or even such to allow an explicit solution of the equations of motion. Some of these systems have the form of classical one-dimensional many-body problems with pair interactions; others are more general. All of them are related to Lie algebras, and in all known cases the property of integrability results from the presence of higher (hidden) symmetries. This review presents from a general and universal viewpoint the results obtained in this field during the last few years. Besides, it contains some new results both of physical and mathematical interest. The main focus is on the one-dimensional models of n particles interacting pairwise via potentials V(q) = g 2v(q) of the following 5 types: v~(q) = q~2, vn(q) = ~~-2sinh2(aq), vui(q) = a2/sin2(aq), Vp,, = a2 ~(aq), vv(q) = q~2+ w2q2. Here ~(q) isthe Weierstrass function, so that the first 3 cases are merely subcases of the fourth. The system characterized by the Toda nearest-neighbor potential, gj exp[— a(q~ — qj+j)], is moreover considered. Various generalizations of these models, naturally suggested by their association with Lie algebras, are also treated. Single orders for (his issue PHYSICS REPORTS (Review Section of Physics Letters) 71, No. 5 (1981) 313-400. Copies of this issue may be obtained at the price given below. All orders should be sent directly to the Publisher. Orders must be accompanied by check. Single issue price Dfl. 40.00, postage included. 0370—1573/81/0000—0000/$17.60 © North-Holland Publishing Company