The Moser-Veselov equation

• Author(s): Cardoso, J. R. ; Leite, F. Silva
• Date: 2001
• Persistent ID: http://hdl.handle.net/10316/11463
• Origin: Estudo Geral - Universidade de Coimbra
• Subject(s): Algebraic Riccati equation; Controllability; Stability; Primary matrix functions
• Document Type: Preprint
• Language: English

Description

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.