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Symplectic integrators offer many advantages for the numerical solution ofHamiltonian differential equations, including bounded energy error and thepreservation of invariant sets. Two of the central Hamiltonian systemsencountered in plasma physics --- the flow of magnetic field lines and theguiding center motion of magnetized charged particles --- resist symplecticintegration by conventional means because the dynamics are most naturallyformulated in non-canonical coordinates, i.e., coordinates lacking the familiar$(q, p)$ partitioning. Recent efforts made progress toward non-canonicalsymplectic integration of these systems by appealing to the variationalintegration framework; however, those integrators were multistep methods andlater found to be numerically unstable due to parasitic mode instabilities.This work eliminates the multistep character and, therefore, the parasitic modeinstabilities via an adaptation of the variational integration formalism thatwe deem ``degenerate variational integration''. Both the magnetic field lineand guiding center Lagrangians are degenerate in the sense that their resultantEuler-Lagrange equations are systems of first-order ODEs. We show thatretaining the same degree of degeneracy when constructing a discrete Lagrangianyields one-step variational integrators preserving a non-canonical symplecticstructure on the original Hamiltonian phase space. The advantages of the newalgorithms are demonstrated via numerical examples, demonstrating superiorstability compared to existing variational integrators for these systems andsuperior qualitative behavior compared to non-conservative algorithms.

Degenerate Variational Integrators for Magnetic Field Line Flow and Guiding Center Trajectories