A Hamiltonian time crystal can emerge when a Noether symmetry is subject to a condition that prevents the energy minimum from being a critical point of the Hamiltonian. A somewhat trivial example is the Schrödinger equation of a harmonic oscillator. The Noether charge for its particle number coincides with the square norm of the wave function, and the energy eigenvalue is a Lagrange multiplier for the condition that the wave function is properly normalized. A more elaborate example is the Gross-Pitaevskii equation that models vortices in a cold atom Bose-Einstein condensate. In an oblate, essentially two dimensional harmonic trap the energy minimum is a topologically protected timecrystalline vortex that rotates around the trap center. Additional examples are constructed using coarse grained Hamiltonian models of closed molecular chains. When knotted, the topology of a chain can support a time crystal. As a physical example, high precision all-atom molecular dynamics is used to analyze an isolated cyclopropane molecule. The simulation reveals that the molecular D3h symmetry becomes spontaneously broken. When the molecule is observed with sufficiently long stroboscopic time steps it appears to rotate like a simple Hamiltonian time crystal. When the length of the stroboscopic time step is decreased the rotational motion becomes increasingly ratcheting and eventually it resembles the back-and-forth oscillations of Sisyphus dynamics. The stroboscopic rotation is entirely due to atomic level oscillatory shape changes, so that cyclopropane is an example of a molecule that can rotate without angular momentum. Finally, the article is concluded with a personal recollection how Frank’s and Betsy’s Stockholm journey started.
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