  We propose and compare several projection methods applied to variational
integrators for degenerate Lagrangian systems, whose Lagrangian is of the form
$L = \vartheta(q) \cdot \dot{q} - H(q)$ and thus linear in velocities. While
previous methods for such systems only work reliably in the case of $\vartheta$
being a linear function of $q$, our methods are long-time stable also for
systems where $\vartheta$ is a nonlinear function of $q$. We analyse the
properties of the resulting algorithms, in particular with respect to the
conservation of energy, momentum maps and symplecticity. In numerical
experiments, we verify the favourable properties of the projected integrators
and demonstrate their excellent long-time fidelity. In particular, we consider
a two-dimensional Lotka-Volterra system, planar point vortices with
position-dependent circulation and guiding centre dynamics.
