Global modified Hamiltonian for constrained
symplectic integrators
Ernst Hairer
Abstract.
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.
Key Words.
Constrained Hamiltonian systems, symplectic integrators,
partitioned Runge-Kutta methods,
generating functions, backward error analysis, modified
Hamiltonian.