  Finite-dimensional non-canonical Hamiltonian systems arise naturally from
Hamilton's principle in phase space. We present a method for deriving
variational integrators that can be applied to perturbed non-canonical
Hamiltonian systems on manifolds based on discretizing this phase-space
variational principle. Relative to the perturbation parameter $\epsilon$, this
type of integrator can take $O(1)$ time steps with arbitrary accuracy in
$\epsilon$ by leveraging the unperturbed dynamics. Moreover, these integrators
are coordinate independent in the sense that their time-advance rules transform
correctly when passing from one phase space coordinate system to another.
