Variational Integrators as General Linear Methods

In this paper we write variational integrators for degenerate Lagrangian systems as general linear methods which turn out to be G-symplectic in nature. Since variational integrators for degenerate Lagrangian systems suffer from parasitic instabilities, we use the G-symplectic general linear method formulation to calculate their parasitic growth parameters, thus enabling us to devise strategies to control numerical instabilities while preserving the underling physical laws of the system. Variational integrators based on trapezoidal and mid-point quadrature rules are considered for degenerate Lagrangian systems resulting in two different classes of general linear methods. We then apply the standard projection technique to project the numerical solution on whatever manifold the exact solution lies on, resulting in energy preservation for long time. The numerical results verify our claims.