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. 1
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