  We develop an index theory for parabolic and collision solutions to the
classical n-body problem and we prove sufficient conditions for the finiteness
of the spectral index valid in a large class of trajectories ending with a
total collapse or expanding with vanishing limiting velocities. Both problems
suffer from a lack of compactness and can be brought in a similar form of a
Lagrangian System on the half time line by a regularising change of coordinates
which preserve the Lagrangian structure. We then introduce a Maslov-type index
which is suitable to capture the asymptotic nature of these trajectories as
half-clinic orbits: by taking into account the underlying Hamiltonian structure
we define the appropriate notion of geometric index for this class of solutions
and we develop the relative index theory.
