Preserving Poisson Structure and Orthogonality in Numerical Integration of Differential Equations

We consider the numerical integration of two types of systems of differential equations. We first consider Hamiltonian systems of differential equations with a Poisson structure. We show that symplectic Runge-Kutta methods preserve this structure when the Poisson tensor is constant. Using nonlinear changes of coordinates this structure can also be preserved for non-constant Poisson tensors, as exemplified on the Euler equations for the free rigid body. We also consider orthogonal flows and the closely related class of isospectral flows. To numerically preserve the orthogonality property we take the approach of formulating an equivalent system of differential-algebraic equations (DAEs) and of integrating the system with a special combination of a particular class of Runge-Kutta methods. This approach requires only matrix-matrix products and can preserve geometric properties of the flow such as reversibility.