from the symmetry considerations, we propose a general recipe for obtaining the many-body Hamiltonian describing a system of interacting spin-S particles. Such systems are characterized by $(2S+1)^{2}-1$ parameters among which there are three components of the magnetization vector and the rest can be treated as the multipole degrees of freedom. These additional parameters are induced in a many-body system by the spin of the structural constituents of matter (particles) and appear at the macroscopic level. The spin degrees of freedom of a particle generate also the SU(2S+1) symmetry of the many-body Hamiltonian in the absence of external fields. As the most intriguing case with a view to physics of ultracold Bose gases, we present a detailed study of the SU(3) symmetric Hamiltonian describing a system of spin-1 particles. Then we apply it to examine the ferromagnetic phase of a weakly interacting Bose gas with condensate in a magnetic field. It is shown that the constructed Hamiltonian modifies the ground state properties and single-particle excitation spectra of the system.

Formulation of the problem and many-body interaction Hamiltonian of internal degrees of freedom
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Consider a many-body system of spin-S particles whose reduced description is performed in terms of the single-particle density 