# Variational principle for cylindrical curves and dynamics of spinning particles in $d=3$ Minkowski space

## TL;DR

This paper develops a variational framework for describing the classical trajectories of spinning particles in 3D Minkowski space, revealing their geometric nature and connection to Poincare group orbits.

## Contribution

It introduces a novel variational principle for cylindrical curves representing spinning particles, including gauge invariance and explicit connection to Poincare group co-orbits.

## Key findings

- Classical paths lie on a circular cylinder in Minkowski space.
- Derived non-Lagrangian equations of motion with a variational interpretation.
- Confirmed states lie on Poincare group co-orbits.

## Abstract

We proceed from the fact that the classical paths of irreducible massive spinning particle lie on a circular cylinder with the time-like axis in Minkowski space. Assuming that all the classical paths on the cylinder are gauge-equivalent, we derive the equations of motion for the cylindrical curves. These equations are non-Lagrangian, but they admit interpretation in terms of the conditional extremum problem for a certain length functional in the class of paths subjected to the constant separation conditions. The unconditional variational principle is obtained after inclusion of constant separation conditions with the Lagrange multipliers into the action. We explicitly verify that the states of the obtained model lie on the co-orbit of the Poincare group. The relationship with the previously known theory is demonstrated.