  Explicit and semi-explicit geometric integration schemes for dissipative
perturbations of Hamiltonian systems are analyzed. The dissipation is
characterized by a small parameter $\epsilon$, and the schemes under study
preserve the symplectic structure in the case $\epsilon=0$. In the case
$0<\epsilon\ll 1$ the energy dissipation rate is shown to be asymptotically
correct by backward error analysis. Theoretical results on monotone decrease of
the modified Hamiltonian function for small enough step sizes are given.
Further, an analysis proving near conservation of relative equilibria for small
enough step sizes is conducted.
  Numerical examples, verifying the analyses, are given for a planar pendulum
and an elastic 3--D pendulum. The results are superior in comparison with a
conventional explicit Runge-Kutta method of the same order.
