  A methodology for solving two-point boundary value problems in phase space
for Hamiltonian systems is presented. Using Hamilton-Jacobi theory in
conjunction with the canonical transformation induced by the phase flow, we
show that the generating functions for this transformation solve any two-point
boundary value problem in phase space. Properties of the generating functions
are exposed, we especially emphasize multiple solutions, singularities,
relations with the state transition matrix and symmetries. Then, we show that
using Hamilton's principal function we are also able to solve two-point
boundary value problems, nevertheless both methodologies have fundamental
differences that we explore. Finally, we present some applications of this
theory. Using the generating functions for the phase flow canonical
transformation we are able to solve the optimal control problem (without an
initial guess), to study phase space structures in Hamiltonian dynamical
systems (periodic orbits, equilibrium points) and classical targeting problems
(this last topic finds its applications in the design of spacecraft formation
trajectories, reconfiguration, formation keeping, etc...).
