Lagrangian Method for Chiral Symmetries

The method of phenomenological nonlinear Lagrangians is described in detail for the case of chiral $\mathrm{SU}\left(2\right)\mathrm{}\times \mathrm{SU}\left(2\right)\mathrm{}$. The complete nonlinear Lagrangian is constructed for a theory of pions, nucleons, and vector and axial-vector mesons. The Lagrangian allows a study of the strong and weak interactions of the particles involved. Relations between masses and coupling constants emerge. A model containing only pions and nucleons is also described. Finally, the way to generalize the method to $\mathrm{SU}\left(3\right)\mathrm{}\times \mathrm{SU}\left(3\right)\mathrm{}$ is indicated.