Space-time symmetry and non-reciprocal transport in dynamic mechanical systems

Active mechanical structures whose parameters are modulated in space and time harbor wave phenomena, such as amplification and non-reciprocal transport, that are forbidden in passive materials. One such phenomenon is parametric resonance, which generates exponentially growing or decaying oscillations when parameters are modulated periodically in time. While parametric resonance of individual oscillators is well understood, identifying the conditions for parametric resonance in systems of coupled oscillators remains challenging. In this talk, we will identify and use the non-Hermitian internal symmetries arising from the real-valued and symplectic nature of classical mechanics to determine these parametric resonance conditions. Upon including external symmetries, we find the conditions for modes to be protected from resonating at certain modulation frequencies where they were expected to resonate in the absence of the external symmetry. In particular, we analyze systems with space-time symmetry where the system remains invariant after a combination of discrete translation in both space and time. For such systems, we identify a combined space-time translation operator that provides more information about the system than the Floquet operator does, and use it to derive conditions for non-reciprocal wave amplification in mechanical metamaterials. Our results establish an exact theoretical framework based on symmetries to engineer non-reciprocal transport in out-of-equilibrium systems.