Broken solutions of homogeneous variational problems

Abstract

A real-valued function L on the tangent bundle of R n gives rise to variational problems as follows: for two points x 0, x 1 in R n and a time interval [0, T] to determine a curve γ: [O,T] → R n, connecting x 0 with x 1 which minimizes ∫ 0 T L( γ( t), gg( t)) dt. We consider the associated Hamiltonian vectorfield on the cotangent bundle. If L is not convex on each fibre then the corresponding Hamiltonian vectorfield is not continuous. For homogeneous L and n = 2 restriction to an energy level gives an essentially three-dimensional vectorfield. In this case we list the possible discontinuities for generic L. Then we observe that there exits an open class of such variational problems, which admit no minimizing solution.