Time-dependent Lagrangians invariant by a vector field

TL;DR

This paper investigates the reduction of non-autonomous Lagrangian systems with symmetries generated by vector fields linked to connections, generalizing time-invariance concepts in classical mechanics.

Contribution

It introduces a framework for reducing non-autonomous systems using symmetries from connections, extending traditional time-invariance reduction methods.

Findings

Develops a reduction method for non-autonomous Lagrangian systems

Generalizes the concept of time-invariance through connection-based symmetries

Provides a geometric approach to symmetry reduction in time-dependent systems

Abstract

We study the reduction of non-autonomous regular Lagrangian systems by symmetries, which are generated by vector fields associated with connections in the configuration bundle of the system $Q\times\real\to\real$. These kind of symmetries generalize the idea of ``time-invariance'' (which corresponds to taking the trivial connection in the above trivial bundle).