Variational problem for Hamiltonian system on so(k, m) Lie-Poisson manifold and dynamics of semiclassical spin

TL;DR

This paper develops a geometric variational framework for Hamiltonian systems on $so(k, m)$ Lie-Poisson manifolds, providing a rigorous mathematical basis for modeling spin as inner angular momentum in semiclassical particles.

Contribution

It introduces a novel geometric construction linking variational problems to Hamiltonian equations on Lie-Poisson manifolds, clarifying the mathematical structure of spin dynamics.

Findings

Derived Hamiltonian equations from variational principles on $so(k, m)$ manifolds.

Identified the manifold as a base of a fiber bundle with a canonical Poisson structure.

Provided a mathematical formulation for spin as inner angular momentum.

Abstract

We describe the procedure for obtaining Hamiltonian equations on a manifold with $so(k, m)$ Lie-Poisson bracket from a variational problem. This implies identification of the manifold with base of a properly constructed fiber bundle embedded as a surface into the phase space with canonical Poisson bracket. Our geometric construction underlies the formalism used for construction of spinning particles in [24-27], and gives precise mathematical formulation of the oldest idea about spin as the "inner angular momentum".