Abstract:

We expand the solutions of linearly coupled Mathieu equations in terms of infinite-continued matrix inversions, and use it to find the modes which diagonalize the dynamical problem. This allows obtaining explicitly the (Floquet–Lyapunov) transformation to coordinates in which the motion is that of decoupled linear oscillators. We use this transformation to solve the Heisenberg equations of the corresponding quantum-mechanical problem, and find the quantum wavefunctions for stable oscillations, expressed in configuration space. The obtained transformation and quantum solutions can be applied to more general linear systems with periodic coefficients (coupled Hill equations, periodically driven parametric oscillators), and to nonlinear systems as a starting point for convenient perturbative treatment of the nonlinearity.