Periodic and Aperiodic Orbits in the Hamiltonian Formulation of a Model Magnetic System

Authors

Charles Kaufman, University of Rhode IslandFollow
Niraj Srivastava, University of Rhode Island
Gerhard Müller, University of Rhode IslandFollow

Document Type

Article

Date of Original Version

1989

Abstract

Classical dynamics and quantum dynamics have influenced each other since the idea of a quantum mechanics originated. Classical dynamics came first, so its influence on quantum theory almost goes without saying. Quantum mechanics grew out of classical mechanics. The converse influence is often referred to in the abstract, but rarely in detail. One finds statements [1] roughly to the effect that the classical theory was developed more fully in order to use it to further elucidate the corresponding quantum dynamics. But specific examples of classical calculations, which were suggested by quantum results or ideas, are not common. One of these rare examples [2] is ‘rotators’ or ‘classical spins,’ and that is the subject considered here.

This study makes contact with optimization theory at two places. First, the spin problem is initially expressed as an application of Hamiltonian dynamics; that is, it is simply an explicit particular example of the principle of least action. In the course of solution, we uncover two qualitatively different types of behavior, viz ‘regular’ and ‘chaotic,’ whose occurrence depends on the value of a (control) parameter. The chaotic solutions, moreover, are not equally chaotic; there is a more-or-less smooth progression into and back out of chaos as the parameter changes. The second contact with control theory then is a question; can the ‘intensity’ of the chaos be quantified, and if so, is there a value of the parameter for which the system is maximally chaotic?

The paper has four sections. Section 2 is a review of classical dynamics, including a description of numerical techniques for distinguishing regular from chaotic motion. Section 3 describes how the quantum mechanical form of a classical dynamics problem is produced. Section 4 discusses the exchange-type interactions relevant to classical spins and presents results of numerical integration for one specific such model.

Citation/Publisher Attribution

Charles Kaufman, Niraj Srivastava and Gerhard Müller. Periodic and aperiodic orbits in the Hamiltonian formulation of a model magnetic system. In Modern Optimal Control, ed. by E.O. Roxin, Marcel Dekker Inc. New York 1989. pp 217-235.