The Moser-Veselov equation

Abstract

We study the orthogonal solutions of the matrix equation XJ-JXT=M, where J is symmetric positive definite and M is skew-symmetric. This equation arises in the discrete version of the dynamics of a rigid body, investigated by Moser and Veselov [15]. We show connections between orthogonal solutions of this equation and solutions of a certain algebraic Riccati equation. This will bring out the symplectic geometry of the Moser-Veselov equation and also reduces most computational issues about solutions to finding invariant subspaces of a certain Hamiltonian matrix. Necessary and sufficient conditions for the existence of orthogonal solutions (and methods to compute them) are presented. Our method is contrasted with the Moser-Veselov approach presented in [15]. We also exhibit explicit solutions of a particular case of the Moser-Veselov equation, which appears associated with the continuous version of the dynamics of a rigid body.