  First-order accurate degenerate variational integration (DVI) was introduced
in C. L. Ellison et. al, Phys. Plasmas 25, 052502 (2018) for systems with a
degenerate Lagrangian, i.e. one in which the velocity-space Hessian is
singular. In this paper we introducing second order accurate DVI schemes, both
with and without non-uniform time stepping. We show that it is not in general
possible to construct a second order scheme with a preserved two-form by
composing a first order scheme with its adjoint, and discuss the conditions
under which such a composition is possible. We build two classes of second
order accurate DVI schemes. We test these second order schemes numerically on
two systems having noncanonical variables, namely the magnetic field line and
guiding center systems. Variational integration for Hamiltonian systems with
nonuniform time steps, in terms of an extended phase space Hamiltonian, is
generalized to noncanonical variables. It is shown that preservation of proper
degeneracy leads to single-step methods without parasitic modes, i.e. to
non-uniform time step DVIs. This extension applies to second order accurate as
well as first order schemes, and can be applied to adapt the time stepping to
an error estimate.
