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We demonstrate that non-convex Lagrangians, as contemplated in the theory oftime crystals, can arise in the effective description of conventional,physically realizable systems. Such embeddings resolve dynamical singularitieswhich arise in the reduced description. Microstructure featuring intervals offixed velocity interrupted by quick resets - ``Sisyphus dynamics'' - is ageneric consequence. In quantum mechanics this microstructure can be blurred,leaving entirely regular behavior.

Regularizations of Time Crystal Dynamics