Obstructions to the Hasse principle for variational problems

A variational problem may be formulated on chain spaces with coefficients in $\mathbb{R}$, $\mathbb{Z}$, and $\mathbb{Z}/n\mathbb{Z}$ for $n\geq 2$. Obstructions to the Hasse principle for variational problems. What are the obstructions to reconstructing the solutions in $\mathbb{Z}$-chains from the corresponding solutions in $\mathbb{R}$-chains and $\mathbb{Z}/n\mathbb{Z}$-chains? This is posed as an open question by analogy with obstructions to the classical Hasse principle in number theory.