A GEOMETRIC APPROACH TO THE SECOND NOETHER’S THEOREM IN TIME-DEPENDENT LAGRANGIAN MECHANICS Jos´ e F. Cari˜ nena † , Jos´ e Fern´ andez-N´ u˜ nez ⋆ and Eduardo Mart´ ınez † † Departamento de F´ ısica Te´ orica, Universidad de Zaragoza, 50009 Zaragoza (Spain) ⋆ Departamento de F´ ısica, Universidad de Oviedo, 33007 Oviedo (Spain) Abstract. We analyse the Second Noether’s Theorem from a geometric viewpoint using the concepts of vector fields and forms along tangent bundle projections 1. Introduction and basic notation Gauge theory has remained since its discovering a fundamental problem in both classical and quantum physics, the very existence of gauge symmetries being a sign of that not all dynamical coordinates are independent [1]. The interest of constrained systems has motivated a lot of papers trying to analyse some still dark points in the theory and many efforts have been devoted to elucidate the behaviour of such systems from a geometric viewpoint (see e.g. [2]). Unfortunately most of progress have been obtained using a time-independent approach, perhaps because the tools of Symplectic Geometry are much more familiar to physicists than those of jet-bundle theory, even in the simplest case of trivial bundles corresponding to classical mechanical systems. On the other hand, the geometrical description of these systems is important not only by its own but because it may be considered as a first step in the geometric study of Classical Field Theory. Anyway, there is no possibility of dealing with the Second Noether Theorem in the framework of time-independent systems because then the arbitrary functions of time arising in the gauge symmetry would have no meaning at all. In a recent paper [3] we have shown that appropriate geometric tools for the study of symmetry in time-dependent systems includes the concepts of vector fields and forms along different bundle projections. These concepts have very seldom been used in physics (see e.g. [4], [5]), but we hope they will be quickly incorporated in the background of theoretical physicists. In particular, the concept of generalised infinitesimal symmetry of a Lagrangian introduced in [3] will be shown to be very useful to give a geometric version of the Second Noether’s theorem as it did for the first Theorem and allows to determine the infinitesimal gauge symmetries of a singular Lagrangian without passing through the Hamiltonian formalism and the invariance of the action as it is usually done. In the geometric approach to time-dependent Lagrangian mechanics the configuration space Q is substituted for the trivial bundle π : R × Q → R and the evolution space for PACS: 02.40.+m, 03.20.+i Typeset by A M S-T E X 1