  The general theoretical ground for the models based on the compact angle
coordinates is presented. It is observed that the proper dependence on compact
coordinates has to be through the group elements and is achieved most naturally
in a discrete-time formulation of the theory. By the construction, the discrete
worldline inlaid by compact coordinates resembles the spin chains of magnetic
systems. As examples, the models based on the groups U(1), $\mathbb{Z}_N$ and
SU(2) are explicitly constructed and their exact energy spectra are obtained.
As the consequence of minima in the spectra, the models exhibit a phase
transition of first-order. The dynamics by U(1) group is attempted to be fitted
to the proposed role for monopoles in the dual Meissner effect of confinement
mechanism.
