Singularities of the Hamiltonian vectorfield in nonautonomous variational problems

Abstract

Variational problems with n degrees of freedom give rise (by Pontriaguine maximum principle) to a Hamiltonian vectorfield in T ∗ R n , that presents singularities (nonsmoothness points) when the Lagrangian is not convex. For one degree of freedom nonautonomous problems of the calculus of variations where the Hamiltonian vectorfield in T ∗ R depends explicitly on the time, we consider the associated autonomous vectorfield in T ∗ R× R and classify its singularities up to an equivalence that takes into account the special role played by the time coordinate, i.e., that respects the foliation of T ∗ R× R into planes of constant time.