Time-dependent kinetic energy metrics for Lagrangians of electromagnetic type

TL;DR

This paper extends previous work on Lagrangian systems by exploring time-dependent kinetic energy metrics, showing they have constant eigenvalues and enable coordinate transformations that partially decouple the system.

Contribution

It introduces a framework for time-dependent kinetic energy metrics in electromagnetic-type Lagrangians, revealing their eigenvalue properties and decoupling capabilities.

Findings

Time-dependent metrics have constant eigenvalues.

Such metrics induce coordinate transformations that partially decouple the system.

Extension of previous results to explicit time-dependent systems.

Abstract

We extend the results obtained in a previous paper about a class of Lagrangian systems which admit alternative kinetic energy metrics to second-order mechanical systems with explicit time-dependence. The main results are that a time-dependent alternative metric will have constant eigenvalues, and will give rise to a time-dependent coordinate transformation which partially decouples the system.