  We show that, when applied to any non-canonical Hamiltonian system, any
integrator that is symplectic for canonical Hamiltonian problems is actually
conjugate symplectic for the non-canonical structure. This result is useful
because it implies that canonically symplectic methods may be successfully
applied to long-time integrations of non-canonical Hamiltonian problems, thus
avoiding the need to construct ad hoc new methods. Numerical results for three
non-canonical Hamiltonian systems demonstrate that (canonically) symplectic
methods have significant advantages in numerical accuracy and near energy
preservation over non-symplectic methods.
