Coupling of Eigenvalues with Applications in Physics and Mechanics
This is a review paper presenting a general theory of coupling of eigenvalues of complex matrices of an arbitrary dimension
depending on real parameters. The cases of weak and strong
coupling are distinguished and their geometric interpretation in
two and three-dimensional spaces is given. General symptotic
formulae for eigenvalue surfaces near diabolic and exceptional
points are presented demonstrating crossing and avoided crossing scenarios. A physical example on propagation of light in a homogeneous non-magnetic crystal illustrates effectiveness and accuracy of the presented theory. As applications in mechanics stability problems for a pendulum with periodically varying length and stabilization effect for a buckled elastic rod by longitudinal vibrations are considered.