Inverse Variational Problem and Symmetry in Action: The Relativistic Third Order Dynamics

TL;DR

This paper develops a method combining symmetry principles and inverse variational analysis to derive a unique third-order Poincaré-invariant equation describing the motion of a relativistic top in three-dimensional space-time.

Contribution

It introduces a unified approach to the inverse variational problem using vector-valued differential forms and identifies a unique relativistic third-order variational equation.

Findings

Derived a covector third-order Poincaré-invariant variational equation

Connected the equation to the motion of a free relativistic top

Established a method for solving invariant inverse problems in relativistic dynamics

Abstract

Tools of the intrinsic analysis on manifolds, helpful in solving the invariant inverse problem of the calculus of variations are being presented comprising a combined approach which consists in the simultaneous imposition of symmetry principles and the inverse variational problem considerations in terms of vector-valued differential forms. In three-dimensional space-time we obtain a unique (covector) third-order Poincar\'e-invariant variational equation, which then is identified with the motion of a free relativistic top in flat three-dimensional space-time.