  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 Poincar{\'e} variational problems.
