2003/07/16 by James D. E. Grant, Grant, James D. E., Emilio Musso +1 · 1 citation
Mathematics · #37K10 #53D20 #58A10 #58A30 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:37K10 #msc:53D20 #msc:58A10 #msc:58A30
paper · pdf · doi:10.48550/arxiv.math/0307216
arxiv created 2003/07/16 · arxiv updated 2009/12/01
In this article we study constrained variational problems in one independent variable defined on the space of integral curves of a Frenet system in a homogeneous space G/H. We prove that if the Lagrangian is G-invariant and coisotropic then the extremal curves can be found by quadratures. Our proof is constructive and relies on the reduction theory for coisotropic optimal control problems. This gives a unified explanation of the integrability of several classical variational problems such as the total squared curvature functional, the projective, conformal and pseudo-conformal arc-length functionals, the Delaunay and the Poincaré variational problems.