Sub-Lagrangians and sub-Hamiltonians on affine bundles

Abstract

In this paper, sub-Lagrangians and sub-Hamiltonians are defined on anchored affine bundles as a natural extension of sub-Riemannian metrics. A duality considered between regular sub-Lagrangians and sub-Hamiltonians, gives the same solution of the Euler–Lagrange and Hamilton equations. Using the Pontryagin maximum principle, we prove that a similar situation of sub-Riemannian minimizers is encountered in this case, i.e., for a positive-definite sub-Lagrangian (sub-Hamiltonian), the locally arc-minimizing curves are either regular ones (as solutions of the Euler–Lagrange and Hamilton equations) or abnormal minimizers.