  We show that a direct shooting method is mathematically equivalent to an
indirect method in the sense of certain first-order conditions. Specific
mathematical formulas pertaining to the equivalence of a direct shooting method
with an indirect method are derived. We also show that a theoretical
equivalence does not necessarily translate to practical equivalence if the
parameterized optimal control problem is simply patched to a nonlinear
programming solver. A mathematical explanation is provided for the successes
and failures of such patched nonlinear programming methods. In order to
generate the correct solution more consistently, the nonlinear programming
solver used in a traditional direct method must be replaced or augmented by a
Hamiltonian programming method. The theoretical results derived in this paper
further strengthen the connections between computational optimal control, deep
learning and automatic differentiation.
