Noether's theorem in time-dependent Lagrangian mechanics (Q1308960)

The authors give an approach to the study of symmetries for time-dependent Lagrangian systems. They consider the geometric framework for the formulation of a Noether-type theorem admitting a converse, when velocity-dependent transformations are allowed. Using the concept of vector fields and differential forms along a map the authors prove the theorem generalizing Noether's results and admitting a converse by considering arbitrary point transformations. They provide a number of geometric constructions giving rise to a concept of the total time derivative operator and to the \(k\)-prolongation of a vector field along \(\pi_{1,0}\) (canonical projection in the jet-bundle). The regarded version of Noether's theorem is illustrated by means of some known examples like Kepler's problem and the 1-dimensional time-dependent harmonic oscillator.