Dynamics of relativistic particle with Lagrangian dependent on acceleration

TL;DR

This paper analyzes the dynamics of relativistic particles with Lagrangians depending on worldline curvature, showing integrability and applying the method to models with maximal proper acceleration, linking constants of motion to mass and spin.

Contribution

It introduces a method to reformulate equations of motion using principal curvatures, demonstrating integrability for arbitrary Lagrangians depending on curvature.

Findings

Equations of motion are integrable for any ${ m L}(k_1)$.

Constants of integration correspond to particle mass and spin.

Model with maximal proper acceleration is consistent and obeys acceleration limits.

Abstract

Models of relativistic particle with Lagrangian ${\cal L}(k_1)$, depending on the curvature of the worldline $k_1$, are considered. By making use of the Frenet basis, the equations of motion are reformulated in terms of the principal curvatures of the worldline. It is shown that for arbitrary Lagrangian function ${\cal L}(k_1)$ these equations are completely integrable, i.e., the principal curvatures are defined by integrals. The constants of integration are the particle mass and its spin. The developed method is applied to the study of a model of relativistic particle with maximal proper acceleration, whose Lagrangian is uniquely determined by a modified form of the invariant relativistic interval. This model gives us an example of a consistent relativistic dynamics obeying the principle of a superiorly limited value of the acceleration, advanced recently.