Stability, complex modes and non-separability in rotating quadratic potentials

TL;DR

This paper analyzes the complex dynamics of particles in rotating quadratic potentials, revealing unstable behaviors, complex modes, and stabilization methods via magnetic field tuning, with implications for non-separable and non-diagonalizable systems.

Contribution

It introduces a complex mode formalism to study unstable rotating quadratic potentials, uncovering rich structures and stabilization mechanisms not previously detailed.

Findings

Unstable systems exhibit complex normal modes.

Dynamics can be stabilized by magnetic field or rotational frequency adjustments.

Non-diagonalizable cases are thoroughly discussed.

Abstract

We examine the dynamics of a particle in a general rotating quadratic potential, not necessarily stable or isotropic, using a general complex mode formalism. The problem is equivalent to that of a charged particle in a quadratic potential in the presence of a uniform magnetic field. It is shown that the unstable system exhibits a rich structure, with complex normal modes as well as non-standard modes of evolution characterized by equations of motion which cannot be decoupled (non-separable cases). It is also shown that in some unstable cases the dynamics can be stabilized by increasing the magnetic field or tuning the rotational frequency, giving rise to dynamical stability or instability windows. The evolution in general non-diagonalizable cases is as well discussed.