Global modified Hamiltonian for constrained symplectic integrators

We prove that the numerical solution of partitioned Runge-Kutta methods applied to constrained Hamiltonian systems (e.g., the Rattle algorithm or the Lobatto IIIA--IIIB pair) is formally equal to the exact solution of a constrained Hamiltonian system with a globally defined modified Hamiltonian. This property is essential for a better understanding of their longtime behaviour. As an illustration, the equations of motion of an unsymmetric top are solved using a parameterization with Euler parameters.