Stability Criteria and Classification of Singularities for Equivariant Lagrangian Subwianifolds

One of the useful methods of mathematical physics is the one arising from symplectic geometry and associating the singularities of Lagrangian submanifolds with the optical caustics, phase transitions, bifurcation patterns, obstacle geometry etc. In this paper we derive the stability criteria for singularities of rquivariant Lagrangian submanifolds with a compact Lie group action determined by a system with symmetry. The recognition problem and classification list for stable (Z#equivariant singularities is proved. We find that the classified stable local models occur as possible realizations for the equilibrium states in the breaking of symmetry and structural phase transitions. Additionally, the connection between two technically different (bv generatine functions, by Morse families) infinitesimal G-stability conditions for equivariant Lagrangian submanifols is studied and an alternative approach is proposed.