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 M n , 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

* : (Xk,Xk+1) - (Xk+uXk+2) (1.3)

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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