Complete Integrability for Lagrangian Dependent on Acceleration in a Space-Time of Constant Curvature

TL;DR

This paper derives the general solutions for particle motion equations with Lagrangians depending on curvature in constant curvature space-times, revealing integrability and interpreting constants as physical properties.

Contribution

It demonstrates complete integrability of equations for arbitrary curvature-dependent Lagrangians in constant curvature space-times and interprets integration constants physically.

Findings

Solutions expressed as integrals for all Lagrangians ${ m f{ extcal L}}(k_1)$

Constants of integration identified as particle mass and spin

Explicit analysis of linear, quadratic, and maximal acceleration models

Abstract

The equations of motion for a Lagrangian ${\cal L}(k_1)$, depending on the curvature $k_1$ of the particle worldline, embedded in a space--time of constant curvature, are considered and reformulated in terms of the principal curvatures. It is shown that for arbitrary Lagrangian function ${\cal L}(k_1)$ the general solution of the motion equations can be obtained by integrals. By analogy with the flat space--time case, the constants of integration are interpreted as the particle mass and its spin. As examples, we completely investigate Lagrangians linear and quadratic in $(k_1)$ and the model of relativistic particle with maximal proper acceleration, in a space--time with constant curvature.