Structure-preserving algorithms for Birkhoffian systems

Generating function representations of the solution mappings of Birkhoffian systems are studied. On the basis of the representations, numerical algorithms which preserve the Birkhoffian symplectic structure of Birkhoffian systems are constructed. A backward error analysis for Birkhoffian-structure-preserving algorithms is presented. Some numerical experiments for the oscillator system show that the new algorithms can simulate the energy dissipation better than the implicit mid-point rule which is well known to be symplectic for Hamiltonian systems of canonical form and does not preserve the Birkhoffian symplectic structure of the damped oscillator system.