Geodesic
Implicit function theorem as a realization of the Lagrange principle. Abnormal points
Sbornik. Mathematics, Tome 191 (2000) no. 1, pp. 1-24 Cet article a éte moissonné depuis la source Math-Net.Ru

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A smooth non-linear map is studied in a neighbourhood of an abnormal (degenerate) point. Inverse function and implicit function theorems are proved. The proof is based on the examination of a family of constrained extremal problems; second-order necessary conditions, which make sense also in the abnormal case, are used in the process. If the point under consideration is normal, then these conditions turn into the classical ones. 
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A. V. Arutyunov. Implicit function theorem as a realization of the Lagrange principle. Abnormal points. Sbornik. Mathematics, Tome 191 (2000) no. 1, pp. 1-24. http://geodesic.mathdoc.fr/item/SM_2000_191_1_a0/

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