  Index theory revealed its outstanding role in the study of periodic orbits of
Hamiltonian systems and the dynamical consequences of this theory are enormous.
Although the index theory in the periodic case is well-established, very few
results are known in the case of homoclinic orbits of Hamiltonian systems.
Moreover, to the authors' knowledge, no results have been yet proved in the
case of heteroclinic and halfclinic (i.e. parametrised by a half-line) orbits.
  Motivated by the importance played by these motions in understanding several
challenging problems in Classical Mechanics, we develop a new index theory and
we prove at once a general spectral flow formula for heteroclinic, homoclinic
and halfclinic trajectories. Finally we show how this index theory can be used
to recover all the (classical) existing results on orbits parametrised by
bounded intervals.
