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.


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
Full Text: DOI