The Moser-Veselov equation
The orthogonal solutions of the matrix equation \[ XJ-JX^{T}=M\tag{1} \] are studied, where \(J\) is symmetric positive definite and \(M\) is skew-symmetric. This Moser-Veselov equation is the discrete version of the Euler-Arnold equation for the motion of the generalized rigid body. It is shown that every orthogonal solution of (1) can be written in the form \(X=(M/2+S)J^{-1}\), where \(S\) is a symmetric solution of the algebraic Riccati equation \[ S^{2}+S(M/2)+(M/2)^{T}S-(M^{2}/4+J^{2})=0. \] By using the Riccati equations theory, existence and uniqueness theorems are obtained for equation (1) and an algorithm for determining a special orthogonal solution is given. Explicit formulae for the solutions of (1) are obtained in the particular case \(J=I\). This case is associated with the continuous version of the dynamics of a rigid body.
scientific article; zbMATH DE number 1880039
scientific article