On the minimal interaction of $\pi$-mesons

The minimal interaction principle is considered for pion-nucleon interactions in the framework of phenomenological Lagrangians that are invariant under the $SU(2)\times SU(2)$ chiral group. The nucleon is regarded as an elementary particle and as a composite particle. It is shown that the geometric approach in the method of the phenomenological Lagrangians enables one to go beyond the purely group approach and that it leads to additional restrictions on the phenomenologicaI constants. The relationship with the higher symmetries is manifested in this approach in an algebraic realization of the chiral symmetry for Weinberg's matrices $X_a(\lambda)$ of the axial-vector coupling.