Breathers in Hamiltonian $\mathcal{P}\mathcal{T}$-symmetric chains of coupled pendula under a resonant periodic force

We derive a Hamiltonian version of the $\mathcal{P}\mathcal{T}$-symmetric discrete nonlinear Schr"{o}dinger equation that describes synchronized dynamics of coupled pendula driven by a periodic movement of their common strings. In the limit of weak coupling between the pendula, we classify the existence and spectral stability of breathers (time-periodic solutions localized in the lattice) supported near one pair of coupled pendula. Orbital stability or instability of breathers is proved in a subset of the existence region.