  We explore the connection between the equations describing Sisyphus dynamics
and the generic Li\'{e}nard type or Li\'{e}nard equation from the viewpoint of
branched Hamiltonians. The former provides the appropriate setting for
classical time crystal being derivable from a higher order Lagrangian. However
it appears the equations of Sisyphus dynamics have a close relation with the
Li\'{e}nard-II equation when expressed in terms of the `velocity' variable.
Another interesting feature of the equations of Sisyphus dynamics is the
appearance of velocity dependent "mass function" in contrast to the more
commonly encountered position dependent mass. The consequences of such mass
functions seem to have connections to cosmological time crystals .
