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| In this talk we propose higher-order variational integrators for a class of degenerate systems described by Lagrangians that are linear in velocities. We analyze the geometry underlying such systems and develop the appropriate theory for variational integration. Our main observation is that the evolution takes place on the primary constraint and the \textquoteleft Hamiltonian' equations of motion can be formulated as an index-1 differential-algebraic system. We also construct variational Runge-Kutta methods and analyze their
properties. The general properties of Runge-Kutta methods depend on the \textquoteleft velocity' part of the Lagrangian. If the \textquoteleft velocity' part is also linear in the position
coordinate, then we show that non-partitioned variational Runge-Kutta methods are equivalent to integration of the corresponding first-order Euler-Lagrange equations, which have the form of a Poisson system with a constant structure matrix, and the classical properties of the Runge-Kutta method are retained. If the \textquoteleft velocity' part
is nonlinear in the position coordinate, we observe a reduction of the order of convergence, which is typical of numerical integration of DAEs. We verify our results through numerical experiments for various dynamical systems. |
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