We consider the possibility that classical dynamical systems display motion in their lowest energy state, forming a time analogue of crystalline spatial order. Challenges facing that idea are identified and overcome. We display arbitrary orbits of an angular variable as lowest-energy trajectories for nonsingular Lagrangian systems. Dynamics within orbits of broken symmetry provide a natural arena for formation of time crystals. We exhibit models of that kind, including a model with traveling density waves.See:Classical Time Crystals
Difficulties around the idea of spontaneous breaking of time translation symmetry in a closed quantum mechanical system are identified, and then overcome in a simple model. The possibility of ordering in imaginary time is also discussed.See: Quantum Time Crystals
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Evidently, something powerful had happened in my brain.FRANK WILCZEK
Simplicity leads to beauty: For it leads, as we've seen, to symmetry, which is an aspect of beauty. As, for that matter, are depth and elegance.
Thus simplicity, properly understood, explains what it is that makes a good explanation deep, elegant, and beautiful.Frank Wilczek