EXTENDED TANGENT BUNDLE FORMALISM FOR TIME-DEPENDENT LAGRANGIAN SYSTEMS

The Lagrangian formalism for time-dependent systems is developed using vector fields defined in the extended tangent bundle T(Q x R). The definition of the time-independent extended Lagrangian function L associated to a time-dependent Lagrangian L is given and then the techniques of symplectic mechanics are used to prove some properties of the equations determing the Euler-Lagrange vector field. The relations between the time-independent and the time-dependent Helmholtz conditions are analyzed from a geometric perspective and, finally, the Noether theorem for the extended Lagrangian L is considered.