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.
Highlights
Quadratic forms in boson operators or generalized coordinates and momenta are a ubiquitous presence in the theoretical description of diverse physical systems
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 unstable system exhibits a rich structure, with several different dynamical regimes as well as some quite remarkable features, includingathe possibility of becoming nonseparable at the boundaries of regions with distinct dynamics, in the sense that the Hamiltonian can no longer be written as a sum of two independent standard or complex modes; in such cases the system will exhibit anomalous evolutions characterized by a set of linear equations which cannot be decoupled and which may lead to coordinates and/or momenta evolving with terms κtet or even κt3 andbthe possibility of achieving dynamical stability in some unstable cases by increasing the magnetic field or tuning the rotational frequency
Summary
Quadratic forms in boson operators or generalized coordinates and momenta are a ubiquitous presence in the theoretical description of diverse physical systems They often arise through the linearization of the equations of motion around a stationary point, as in the case of the random-phase approximationRPA ͓1,2͔, providing a basic tractable scenario. The unstable system exhibits a rich structure, with several different dynamical regimes as well as some quite remarkable features, includingathe possibility of becoming nonseparable at the boundaries of regions with distinct dynamics, in the sense that the Hamiltonian can no longer be written as a sum of two independent standard or complex modes; in such cases the system will exhibit anomalous evolutions characterized by a set of linear equations which cannot be decoupled and which may lead to coordinates and/or momenta evolving with terms κtet or even κt andbthe possibility of achieving dynamical stability in some unstable cases by increasing the magnetic field or tuning the rotational frequency.
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