Hydrogen is one of the few molecules that has been incarcerated in the molecular cage of C
60
to form the endohedral supramolecular complex H
2
@C
60
. In this confinement, hydrogen acquires new properties. Its translation motion, within the C
60
cavity, becomes quantized, is correlated with its rotation and breaks inversion symmetry that induces infrared (IR) activity of H
2
. We apply IR spectroscopy to study the dynamics of hydrogen isotopologues H
2
, D
2
and HD incarcerated in C
60
. The translation and rotation modes appear as side bands to the hydrogen vibration mode in the mid-IR part of the absorption spectrum. Because of the large mass difference of hydrogen and C
60
and the high symmetry of C
60
the problem is almost identical to a vibrating rotor moving in a three-dimensional spherical potential. We derive potential, rotation, vibration and dipole moment parameters from the analysis of the IR absorption spectra. Our results were used to derive the parameters of a pairwise additive five-dimensional potential energy surface for H
2
@C
60
. The same parameters were used to predict H
2
energies inside C
70
. We compare the predicted energies and the low-temperature IR absorption spectra of H
2
@C
70
.