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This paper is concerned with equilibrium configurations of one-dimensionalparticle system with non-convex nearest-neighbour and next-to-nearest-neighbourinteractions and its passage to the continuum. The goal is to derivecompactness results for a $\Gamma$-development of the energy with the noveltythat external forces are allowed. In particular, the forces may depend onLagrangian or Eulerian coordinates. Our result is based on a new technique forderiving compactness results which are required for calculating the first-order$\Gamma$-limit: instead of comparing a configuration of $n$ atoms to a globalminimizer of the $\Gamma$-limit, we compare the configuration to a minimizer insome subclass of functions which in some sense are "close to" theconfiguration. This new technique is required due to the additional presence offorces with non-convex potentials. The paper is complemented with the study ofthe minimizers of the $\Gamma$-limit.

External forces in the continuum limit of discrete systems with non-convex interaction potentials: Compactness for a $\Gamma$-development