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.
Previous Page Next Page