Test Particles with Acceleration-Dependent Lagrangian

TL;DR

This paper investigates classical test particles with acceleration-dependent Lagrangians in electromagnetic and gravitational fields, exploring their trajectories, symmetries, and potential issues like run-away solutions and unphysical energies.

Contribution

It introduces a framework for analyzing acceleration-dependent Lagrangians, applying it to models with bounded acceleration and pseudo-acceleration, and discusses their physical implications.

Findings

Models can impose upper bounds on acceleration and pseudo-acceleration.

Unwanted features such as run-away solutions and unphysical energies may occur.

Simple field choices reveal interesting properties and potential issues.

Abstract

We consider a classical test particle subject to electromagnetic and gravitational fields, described by a Lagrangian depending on the acceleration and on a fundamental length. We associate to the particle a moving local reference frame and we study its trajectory in the principal fibre bundle of all the Lorentz frames. We discuss in this framework the general form of the Lagrange equations and the connection between symmetries and conservation laws (Noether theorem). We apply these results to a model, already discussed by other authors, which implies an upper bound to the proper acceleration and to another new model in which a similar quantity, called ``pseudo-acceleration'', is bounded. With some simple choices of the fields, we illustrate some other interesting properties of the models and we show that unwanted features may appear, as unstable run-away solutions and unphysical values of the energy-momentum or of the velocity.