Projected Variational Integrators for Degenerate Lagrangian Systems

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.