Some remarks on numerical methods for second order differential equations on the orthogonal matrix group

In the last years several numerical methods have been developed to integrate matrix differential equations which preserve certain features of the theoretical solution such as orthogonality, eigenvalues, first integrals, etc. In this paper we approach the numerical solution of a second order matrix differential system whose solution evolves on the Lie-group of the orthogonal matrices On . We study the orthogonality properties of classical Runge Kutta Nyström methods and non standard numerical procedures for second order ordinary differential equations.