This work is devoted to show the interest of polar coordinates in the description
of some unitary irreducible representations (or uir’s) of the SO(2, 2) group
where the support space are functions on the three dimensional pseudosphere
H2,2R . We will show that the differential equations associated to such uir’s can
be interpreted as quantum systems including centrifugal terms; in our case
these equations lead to one-dimensional Pöschl-Teller systems. The solutions
to these equations are computed and the uir’s are characterized in terms of polar
coordinates. We will also discuss briefly the more standard pseudospherical
coordinates onH2,2 R in order to appreciate some of the differences. We will consider
as well the (maximally superintegrable) free classical systems defined on
the real pseudosphere H2,2 R symmetric under SO(2, 2). The constants of motion
are found and they are applied to find some bounded (therefore periodic) and unbounded orbits also in terms of polar coordinates.
