Regularization of singular time-dependent Lagrangian systems

Abstract

One approach to studying the dynamics of a singular Lagrangian system is to attempt to regularize it, that is, to find an equivalent and regular system. In the case of time-independent singular Lagrangians, an approach due to \textit{A. Ibort} and \textit{J. Mar\'in-Solano} is to use the coisotropic embedding theorem proved by \textit{M.J. Gotay} which states that any pre-symplectic manifold can be coisotropically embedded in a symplectic manifold. In this paper, we revisit these results and provide an alternative approach, also based on the coisotropic embedding theorem, that employs the Tulczyjew isomorphism and almost product structures, and allows for a slight generalization of the construction. In this revision, we also prove uniqueness of the Lagrangian regularization to first order. Furthermore, we extend our methodology to the case of time-dependent singular Lagrangians.