# What's the difference between a time crystal and a system undergoing periodic motion?

What's the difference between a time crystal and a system undergoing periodic motion? My understanding of a crystal is that it is a rigid body with a spatially periodic structure. Is any system undergoing periodic motion a time crystal, or is there some other requirement that makes it special? In other words, what (if anything) excludes the ideal simple harmonic oscillator from the category of time crystal?

Similarly, a time crystal is a system which is subject to a periodic driving with period $T$, but which however does not oscillate with the same period but shows oscillations which a different period $T'=kT$ with $k>1$ integer, i.e., which spontaneously breaks the (discrete) time translation symmetry.
• Naive (and redundant) question but just to confirm, a system with time-independent Hamiltonian that shows oscillations with some time-period $T>0$ still won't be a time-crystal as long as there is no spontaneous symmetry breaking of time-translational symmetry, right?