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First-order accurate degenerate variational integration (DVI) was introducedin C. L. Ellison et. al, Phys. Plasmas 25, 052502 (2018) for systems with adegenerate Lagrangian, i.e. one in which the velocity-space Hessian issingular. In this paper we introducing second order accurate DVI schemes, bothwith and without non-uniform time stepping. We show that it is not in generalpossible to construct a second order scheme with a preserved two-form bycomposing a first order scheme with its adjoint, and discuss the conditionsunder which such a composition is possible. We build two classes of secondorder accurate DVI schemes. We test these second order schemes numerically ontwo systems having noncanonical variables, namely the magnetic field line andguiding center systems. Variational integration for Hamiltonian systems withnonuniform time steps, in terms of an extended phase space Hamiltonian, isgeneralized to noncanonical variables. It is shown that preservation of properdegeneracy leads to single-step methods without parasitic modes, i.e. tonon-uniform time step DVIs. This extension applies to second order accurate aswell as first order schemes, and can be applied to adapt the time stepping toan error estimate.

Improved accuracy in degenerate variational integrators for guiding center and magnetic field line flow