Weyl points in ball-and-spring mechanical systems

Degeneracy points of parameter-dependent Hermitian matrices play a fundamental role in quantum physics, as illustrated by the concept of the Berry phase in quantum dynamics, the Weyl semimetals in condensed-matter physics, and the robust ground-state degeneracies in topologically ordered quantum systems. Here, we construct simple ball-and-spring mechanical systems, whose eigenfrequency degeneracies mimic the behavior of degeneracy points of electronic band structures. These classical-mechanical arrangements can be viewed as ``de-quantized'' versions of Weyl Josephson circuits, i.e., superconducting nanostructures proposed recently to mimic band-structure effects of Weyl semimetals. In the mechanical setups we study, we identify degeneracy patterns beyond simple Weyl points, including the chirality flip effect and a quadratic degeneracy point. Our theoretical work is a step toward simple and illustrative table-top experiments exploring topological and differential geometrical aspects of physics.