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We explore the connection between the equations describing Sisyphus dynamicsand the generic Li\'{e}nard type or Li\'{e}nard equation from the viewpoint ofbranched Hamiltonians. The former provides the appropriate setting forclassical time crystal being derivable from a higher order Lagrangian. Howeverit appears the equations of Sisyphus dynamics have a close relation with theLi\'{e}nard-II equation when expressed in terms of the `velocity' variable.Another interesting feature of the equations of Sisyphus dynamics is theappearance of velocity dependent "mass function" in contrast to the morecommonly encountered position dependent mass. The consequences of such massfunctions seem to have connections to cosmological time crystals .

Construction of the classical time crystal Lagrangians from Sisyphus dynamics and duality description with the Li\'enard type equation