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The Maxwell set in the generalized Dido problem. (English) Zbl 1144.53044

Sb. Math. 197, No. 4, 595-621 (2006); translation from Mat. Sb. 197, No. 4, 123-150 (2006).
This paper is one in a series by the author dealing with the so-called generalized Dido problem. This is a model for the nilpotent sub-Riemannian problem with growth vector \((2,3,5)\). It is known that normal geodesics can cease to be optimal for two reasons: either different geodesics of equal length intersect at a given point (Maxwell points), or a family of geodesics has an envelope (conjugate points). The series of papers concludes with a method to find the intersections of geodesics with the hypersurfaces of Maxwell points corresponding to the symmetry group preserving time along the geodesics. In the present paper, a general description of the Maxwell strata corresponding to the group of discrete symmetries of the exponential map is obtained. The invariant and geometric meaning of these strata is clarified.

MSC:

53C17 Sub-Riemannian geometry
53C22 Geodesics in global differential geometry
49K15 Optimality conditions for problems involving ordinary differential equations
17B66 Lie algebras of vector fields and related (super) algebras
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