On the exact discrete Lagrangian function for variational integrators: theory and applications
This theoretical work provides a foundation for analyzing variational integrators, which is important for numerical simulations in mechanics and physics.
The paper provides a rigorous construction of the exact discrete Lagrangian for variational integrators within the general framework of Lie groupoids and algebroids, enabling error analysis between exact and discrete trajectories.
In this paper, we will give a rigorous construction of the exact discrete Lagrangian formulation associated to a continuous Lagrangian problem. Moreover, we work in the setting of Lie groupoids and Lie algebroids which is enough general to simultaneously cover several cases of interest in discrete and continuous descriptions as, for instance, Euler-Lagrange equations, Euler-Poincaré equations, Lagrange-Poincaré equations... The construction of an exact discrete Lagrangian is of considerable interest for the analysis of the error between an exact trajectory and the discrete trajectory derived by a variational integrator.