  Symplectic integrators offer many advantages for the numerical solution of
Hamiltonian differential equations, including bounded energy error and the
preservation of invariant sets. Two of the central Hamiltonian systems
encountered in plasma physics --- the flow of magnetic field lines and the
guiding center motion of magnetized charged particles --- resist symplectic
integration by conventional means because the dynamics are most naturally
formulated in non-canonical coordinates, i.e., coordinates lacking the familiar
$(q, p)$ partitioning. Recent efforts made progress toward non-canonical
symplectic integration of these systems by appealing to the variational
integration framework; however, those integrators were multistep methods and
later found to be numerically unstable due to parasitic mode instabilities.
This work eliminates the multistep character and, therefore, the parasitic mode
instabilities via an adaptation of the variational integration formalism that
we deem ``degenerate variational integration''. Both the magnetic field line
and guiding center Lagrangians are degenerate in the sense that their resultant
Euler-Lagrange equations are systems of first-order ODEs. We show that
retaining the same degree of degeneracy when constructing a discrete Lagrangian
yields one-step variational integrators preserving a non-canonical symplectic
structure on the original Hamiltonian phase space. The advantages of the new
algorithms are demonstrated via numerical examples, demonstrating superior
stability compared to existing variational integrators for these systems and
superior qualitative behavior compared to non-conservative algorithms.
