Table Of ContentLoop Groups, Discrete Versions
of Some Classical Integrable Systems,
and Rank 2 Extensions
Recent Titles in This Series
479 Percy Deift, Luen-Chau Li, and Carlos Tomei, Loop groups, discrete versions of some
classical integrable systems, and rank 2 extensions, 1992
478 Henry C. Wente, Constant mean curvature immersions of Enneper type, 1992
477 George E. Andrews, Bruce C. Berndt, Lisa Jacobsen, and Robert L. Lamphere, The
continued fractions found in the unorganized portions of Ramanujan's notebooks, 1992
476 Thomas C. Hales, The subregular germ of orbital integrals, 1992
475 Kazuaki Taira, On the existence of Feller semigroups with boundary conditions, 1992
474 Francisco Gonzalez-Acuna and Wilbur C. Whitten, Imbeddings of three-manifold groups,
1992
473 Ian Anderson and Gerard Thompson, The inverse problem of the calculus of variations
for ordinary differential equations, 1992
472 Stephen W. Semmes, A generalization of riemann mappings and geometric structures on
a space of domains in Cn, 1992
471 Michael L. Mihalik and Steven T. Tschantz, Semistability of amalgamated products and
HNN-extensions, 1992
470 Daniel K. Nakano, Projective modules over Lie algebras of Cartan type, 1992
469 Dennis A. Hejhal, Eigenvalues of the Laplacian for Hecke triangle groups, 1992
468 Roger Kraft, Intersections of thick Cantor sets, 1992
467 Randolph James Schilling, Neumann systems for the algebraic AKNS problem, 1992
466 Shari A. Prevost, Vertex algebras and integral bases for the enveloping algebras of affine
Lie algebras, 1992
465 Steven Zelditch, Selberg trace formulae and equidistribution theorems for closed
geodesies and Laplace eigenfunctions: finite area surfaces, 1992
464 John Fay, Kernel functions, analytic torsion, and moduli spaces, 1992
463 Bruce Reznick, Sums of even powers of real linear forms, 1992
462 Toshiyuki Kobayashi, Singular unitary representations and discrete series for indefinite
Stiefel manifolds U(p,q;F)/U{p - m,q;F) 1992
9
461 Andrew Kustin and Bernd Ulrich, A family of complexes associated to an almost
alternating map, with application to residual intersections, 1992
460 Victor Reiner, Quotients of coxeter complexes and P-partitions, 1992
459 Jonathan Arazy and Yaakov Friedman, Contractive projections in Q, 1992
458 Charles A. Akemann and Joel Anderson, Lyapunov theorems for operator algebras, 1991
457 Norihiko Minami, Multiplicative homology operations and transfer, 1991
456 Michal Misiurewicz and Zbigniew Nitecki, Combinatorial patterns for maps of the
interval, 1991
455 Mark G. Davidson, Thomas J. Enright and Ronald J. Stanke, Differential operators and
highest weight representations, 1991
454 Donald A. Dawson and Edwin A. Perkins, Historical processes, 1991
453 Alfred S. Cavaretta, Wolfgang Dahmen, and Charles A. Micchelli, Stationary
subdivision, 1991
452 Brian S. Thomson, Derivates of interval functions, 1991
451 RolfSchdn, Effective algebraic topology, 1991
450 Ernst Dieterich, Solution of a non-domestic tame classification problem from integral
representation theory of finite groups (A = RC$, v(3) = 4), 1991
449 Michael Slack, A classification theorem for homotopy commutative H-spaces with
finitely generated mod 2 cohomology rings, 1991
448 Norman Levenberg and Hiroshi Yamaguchi, The metric induced by the Robin function,
1991
(See the AMS catalog for earlier titles)
MEMOIRS
-LIT A f the
0
American Mathematical Society
Number 479
Loop Groups, Discrete Versions
of Some Classical Integrable Systems,
and Rank 2 Extensions
Percy Deift
Luen-Chau Li
Carlos Tomei
November 1992 • Volume 100 • Number 479 (second of 4 numbers) • ISSN 0065-9266
American Mathematical Society
Providence, Rhode Island
1991 Mathematics Subject Classification.
Primary 34, 70, 15.
Library of Congress Cataloging-in-Publication Data
Deift, Percy, 1945-
Loop groups, discrete versions of some classical integrable systems, and rank 2 extensions/Percy
A. Deift, Luen-Chua Li, Carlos Tomei.
p. cm. - (Memoirs of the American Mathematical Society, ISSN 0065-9266; no. 479)
Includes bibliographical references.
ISBN 0-8218-2540-2
1. Hamiltonian systems. 2. Loops (Group theory) I. Li, Luen-Chau, 1954- . II. Tomei, Car
los. III. Title. IV. Series.
QA3.A57 no. 479
[QA614.83]
510s-dc20 92-28571
[514'.74] CIP
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Table of Contents
Chapter 1. Introduction 1
Chapter 2. The discrete Euler-Arnold equation (I) 12
Chapter 3. The discrete Euler-Arnold equation (II) 42
Chapter 4. Billiards in an elliptical region 59
Chapter 5. Loop groups and rank 2 extensions 83
Appendix. Classical /^-matrix theory 93
Bibliography 99
Abstract
The authors show how to interpret recent results of Moser and Veselov on discrete
versions of a class of classical integrable systems, in terms of a loop group framework. In
this framework the discrete systems appear as time-one maps of integrable Hamiltonian
flows. Earlier results of Moser on isospectral deformations of rank 2 extensions of a fixed
matrix, can also be incorporated into their scheme.
Key words: Hamiltonian mechanics, integrable systems, loop groups, 7?-matrices, discrete
Euler-Arnold equation, billiard map, rank 2 extensions.
Acknowledgements
The work of the first author was supported in part by NSF Grant DMS-9001857. The
work of the second author was supported in part by NSF Grant DMS-8704097. The
third author acknowledges the support of CNP, Brazil, and the warm hospitality of
q
the Department of Mathematics at Yale University. The first author would also like to
acknowledge the support of the Institute for Advanced Study, Princeton, NJ, where part
of this work was completed.
vi 1
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1. Introduction
In a recent paper, Moser and Veselov [MV] considered a class of discrete systems which
arise as the Euler-Lagrange equations for a formal sum
s= ]Tx(x*,x* i), (i.i)
+
where the Xk are points on a manifold Afn, £(•,•) is a function on Q2n = Mn x Mn, and
k £ Z plays the role of the discrete time. As in the analogous continuum problem
S = J L(q,q)dt , (1.2)
one introduces an associated symplectic structure (see [V], [MV]), and the Euler-Lagrange
equations give rise to a mapping
* : (X,X ) -> (X X ) (1.3)
k k+1 k+u k+2
which is symplectic with respect to the structure. Appropriate choices of L and Mn
lead to a wide variety of dynamical systems with many remarkable properties and also of
considerable mathematical and physical interest (see [MV] and the references therein).
Of particular interest in [MV] is the case Mn = 0(N), n = N(N - l)/2, and
L(X,Y) = tr XJYT , (1.4)
where J is a positive symmetric matrix which may be taken to be diagonal without loss
of generality. This problem, introduced in [V], converges in the continuum limit to the
force-free motion of a rigid body as generalized by Arnold to arbitrary dimensions. The
remarkable discovery in [MV] is that the Euler-Lagrange equations for (1.4) can be solved
by a QR-type algorithm. Recall that the classical QR-algorithm for diagonalizing matrices
(see, for example, [Wi]) proceeds by factoring a real invertible matrix M (uniquely) into
a product M = QR of an orthogonal matrix Q and an upper triangular matrix R with
positive diagonal entries. The basic step in the algorithm consists in mapping
Received by editor January 31, 1991.
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