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Abstract. We study the Wigner caustic on shell of a Lagrangian submanifold L of affine symplectic space. We present the physical motivation for studying singularities of the Wigner caustic on shell and present its mathematical definition in terms of a generating family. Because such a generating family is an odd deformation of an odd function, we study simple singularities in the category of odd functions and their odd versal deformations, applying these results to classify the singularities of the Wigner caustic on shell, interpreting these singularities in terms of the local geometry of L.

We study the global centre symmetry set (GCS) of a smooth closed submanifold M m ⊂ R n , n ≤ 2m. The GCS includes both the centre symmetry set defined by Janeczko (Geometria Dedicata 60:9-16, 1996) and the Wigner caustic defined by Berry (Philos Trans R Soc Lond A 287:237-271, 1977) . The definition of GCS(M) uses the concept of an affinedp i ∧ dq i ), we present generating families for singularities of E λ (L) and prove that the caustic of any simple stable Lagrangian singularity in a 4m-dimensional Lagrangian fibre bundle is realizable as the germ of an affine equidistant of some L ⊂ R 2m . We characterize the criminant part of GCS(L) in terms of bitangent hyperplanes to L. Then, after presenting the appropriate equivalence relation to be used in this Lagrangian case, we classify the affine-Lagrangian stable singularities of GCS(L). In particular we show that, already for a smooth closed convex curve L ⊂ R 2 , many singularities of GCS(L) which are affine stable are not affine-Lagrangian stable.

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Abstract. Using standard methods for studying singularities of projections and of contacts, we classify the stable singularities of affine λ-equidistants of n-dimensional closed submanifolds of R q , for q ≤ 2n, whenever (2n, q) is a pair of nice dimensions [12].