  In this note we study two index questions. In the first we establish the
relationship between the Morse indices of the free time action functional and
the fixed time action functional. The second is related to Rabinowitz Floer
homology. Our index computations are based on a correction term which is
defined as follows: around a non-degenerate Hamiltonian orbit lying in a fixed
energy level a well-known theorem says that one can find a whole cylinder of
orbits parametrized by the energy. The correction term is determined by whether
the periods of the orbits are increasing or decreasing as one moves up the
orbit cylinder. We also provide an example to show that, even above the
Ma\~n\'e critical value, the periods may be increasing thus producing a jump in
the Morse index of the free time action functional in relation to the Morse
index of the fixed time action functional.
