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

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