Differential geometric mechanisms in Ostrohrads'kyj relativistic spherical top dynamics

TL;DR

This paper applies differential geometric tools to analyze the variational structure of a third-order relativistic top dynamics, revealing the non-existence of a global Lagrangian and connecting to spinning particle models.

Contribution

It introduces a combined symmetry and inverse variational approach using differential forms for relativistic top dynamics, and discusses the Hamiltonian perspective.

Findings

No global Lagrangian exists for the model.

A transition algorithm between autonomous and parametric variational problems.

Connection to quasi-classical spinning particle models.

Abstract

Some intrinsic tools from the formal theory of variational equations are being demonstrated at work in application to one concrete example of the third-order evolution equation of free relativistic top in three-dimensional space-time. The main goal is to introduce a combined approach consisting in the simultaneous utilization of symmetry principles along with the inverse variational problem considerations in terms of vector-valued differential forms. Next, some simple algorithm of transition between the autonomous variational problem and the variational problem in parametric form is established. The example definitely solved shows no-existence of a globally and intrinsically defined Lagrangian for the Poincar\'e-invariant and well defined unique variational equation in the case in hand. Hamiltonian counterpart is briefly discussed in terms of Poisson bracket. The model appears to provide a generalized canonical description of the quasi-classical spinning particle governed by the Mathisson-Papapetrou equations in flat space-time.