Conservative methods for ordinary differential equations on quadratic groups

In recent years several numerical methods have been developed to integrate matrix differential systems of ODEs whose solutions remain on a certain Lie group throughout the evolution. In this paper it is described a class of methods, based on the Cayley transform, which can be used to solve orthogonal differential systems and which can be extended to the class of quadratic groups including the symplectic and Lorentz group. This approach also applies to ODEs on the Stiefel manifold. Furthermore these methods can solve important problems, such as isospectral flows, the calculation of Lyapunov exponents of dynamical systems, Hamiltonian differential systems and the orthogonal Procustes problem.