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A note on singular Lagrangian submanifolds. (English) Zbl 0593.58010

Summary: Classification of singular lagrangian submanifolds which appear as images of a regular one under a symplectic relation, is considered from the point of view of standard singularity theory. The classification is carried out in small dimensions and restricted to special types of symplectic objects. Normal forms for singular pull-backs and pushforwards are given using an appropriate symplectic equivalence group. It is shown that the general classification problem reduces to the classification problem for appropriate mapping diagrams. An approach to the classical theories of phase transition is given based on the geometry of singular lagrangian images. The variational open swallowtails and regularly intersecting pairs of holonomic components are resolved using an appropriate reduction relation. Examples are given of singularities encountered in physics.

MSC:

58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
57R45 Singularities of differentiable mappings in differential topology
37N99 Applications of dynamical systems
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