Bott periodicity from the point of view of an $n$-dimensional Dirichlet functional

The paper investigates topological effects associated with an $n$-dimensional Dirichlet functional on spaces of representations of disks with fixed boundaries in compact Lie groups $U(n)$, $O(n)$ or $S_p(n)$. It turns out that the classical Bott periodicity arises naturally when one considers the set of points at which the Dirichlet functional attains an absolute minimum, and the periodicity isomorphism is obtained using this approach “in one step” and not in several steps as it was the case when the one-dimensional action functional on the space of loops was used.