The Moser–Veselov equation Original Research Article

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 (Commun. Math. Phys. 139 (1991) 217). 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 (Commun. Math. Phys. 139 (1991) 217). 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.