Regularization of singular time-dependent Lagrangian systems

TL;DR

This paper presents a new approach to regularizing singular time-dependent Lagrangian systems using coisotropic embeddings, Tulczyjew isomorphism, and almost product structures, extending previous methods and proving regularization uniqueness.

Contribution

It introduces an alternative coisotropic embedding method for regularizing time-dependent singular Lagrangians, generalizing prior approaches and establishing regularization uniqueness.

Findings

Extended regularization methodology to time-dependent systems

Proved first-order uniqueness of the Lagrangian regularization

Provided a generalized construction using Tulczyjew isomorphism

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.