  This article studies the existence of long-time solutions to the Hamiltonian
boundary value problem, and their consistent numerical approximation. Such a
boundary value problem is, for example, common in Molecular Dynamics, where one
aims at finding a dynamic trajectory that joins a given initial state with a
final one, with the evolution being governed by classical (Hamiltonian)
dynamics. The setting considered here is sufficiently general so that long time
transition trajectories connecting two configurations can be included, provided
the total energy $E$ is chosen suitably. In particular, the formulation
presented here can be used to detect transition paths between two stable basins
and thus to prove the existence of long-time trajectories. The starting point
is the formulation of the equation of motion of classical mechanics in the
framework of Jacobi's principle; a curve shortening procedure inspired by
Birkhoff's method is then developed to find geodesic solutions. This approach
can be viewed as a string method.
