Hamiltonian lifts and optimal trajectories

The Pontryagin maximum principle says that, if γ is a boundary trajectory of a control system Σ, then γ is a projection of a particular kind of trajectory-a Hamiltonian-minimizing trajectory-of another kind of system, the Hamiltonian lift of Σ. This point of view is useful to prove regularity theorems for optimal trajectories. The author illustrates this by describing some of these theorems, and proving one of them in detail